The medieval authorities clarified that "every woman who makes a vow," etc., is not a condition within the vow, but rather a reason why the Torah placed the woman under his authority regarding the annulment of vows. And Rashba wrote that this right of annulment exists even if the woman explicitly said that she was vowing not subject to his will, because most women want this. If so, it makes sense that nowadays the law would not be different, because in vows that concern matters between him and her, most women presumably want to have an opening for annulment (men would want that too, but the Torah did not grant them that authority), and even if from time to time in the course of a serious quarrel they do not want it, the Torah granted the husband the annulment right based on the majority view.

That is why I cited Tosafot, from whom it seems that Rav Pinchas’s reasoning is indeed a logical argument.

Do Tosafot agree with Rashba’s position?

Aharon,
In annulment of vows there are various limitations, such as that one must say it specifically in a certain formula, or that he must be able to hear the vow (excluding the wife of a deaf-mute), etc. These limitations show that this is a law innovated by the Torah and not a condition stipulated by women, because if it were a condition, the formula and the hearing would not be relevant (see Ran on Nedarim 73b).
What is the relationship between the law of annulment of vows and the principle that "every woman who makes a vow," etc.?
The Torah innovated that since women vow subject to their husbands’ will, they are placed under the husband’s authority for annulment. That is what emerges from Rashba and the Ran cited above. That also seems to be the meaning of the Rosh in Responsa, section 10, where he called "every woman who makes a vow" a "primary reason," and explained there that a primary reason does not depend on the view of Rabbi Shimon, who derives law from the reason for the verse, and therefore laws may be inferred from it. And there is no reason to think that Tosafot would disagree with this principle.
(If not for the words of those medieval authorities, perhaps one could have said that "every woman who makes a vow" is not a rule about the woman’s intention and is not the reason for the law, but rather a law innovated by the Torah that with women’s vows there is a condition dependent on the husband’s will—not because they themselves make it dependent on the husband’s will, but because the Torah makes their vows depend on him.)

As Aharon wrote, Tosafot do not seem to indicate that (rather, that it is an actual condition).
By the way, I once wrote your kind of explanation regarding the rule that whoever betroths does so subject to the rabbis’ will and they can dissolve the betrothal. Usually people understand it as a condition imposed by the husband ("according to the law of Moses and Israel"), whereas the plain sense is that this is a condition imposed by the Torah (similar to the principle that the religious court may stipulate on behalf of others).
Rashba too linked this to the view of most women, so it is not clear that this is not a condition. It is an implicit social condition of society as a whole, and therefore one who vows presumably vows that way. But if it were to happen that a situation arose in which most women no longer vowed that way, it seems that Rashba too would agree. According to Aharon, that is the situation today. Though his claim in general requires examination.
Still, as I wrote, regarding vows of self-affliction and matters between him and her, it is certainly plausible that even today this is so—except that perhaps it is mutual. And by the way, even without the innovation of annulment of vows, if there is a stipulation then it certainly takes effect, so if there is a strong presumption that this is how the woman vowed, then it applies according to all views. We have never found any innovation that a woman cannot make a vow on condition. And in the laws of conditions it is explained that in matters where her mind and the mind of everyone are aligned, we do not require all the formal laws of conditions.
One more remark: the Rosh, who wrote that this is a primary reason not dependent on the dispute regarding the reason for the verse, is probably following his own approach as I explained above (he writes similarly about repealing enactments when their reason has lapsed).

1. I do not know whether Tosafot do not imply this; I did not check, but I would not be surprised if the Ran too, who wrote fairly clearly that it is not a condition, would explain here as Tosafot do.
2. Actually, that is also how I understood "whoever betroths does so subject to the rabbis’ will" (though I have not seen a medieval authority who understood it that way). In vows it is an even clearer argument, because here the Torah explicitly granted the husband the right of annulment.
3. Regarding Rashba, you wrote: "It is an implicit social condition of society as a whole, and therefore one who vows presumably vows that way"—but Rashba wrote this even if she says that she is not vowing subject to her husband’s will (as I understand it: she says so even at the time of the vow (*))
4. Indeed, according to Rashba, if most women did not want this, then perhaps the law would change. But in my humble opinion it does not really depend on an actual, conscious desire at the time of the vow—for it is obvious that even in the time of the Talmud, a woman who forbade her husband marital relations in the middle of a quarrel did so deliberately in order to punish him, not in order that he annul it for her. So what does it mean that she vowed subject to his will? Apparently somewhere in the subconscious there is some reliance on there being a way out and escape once they reconcile. (Such a reasoning is impossible to test empirically, so it is not clear that it changes over time. On the one hand this is a problematic argument because it is not empirical but intuitive, yet on the other hand, how else can one explain the power of annulment even during a quarrel?)
5. It is clear that a woman can stipulate, but the halakhic limitations on the formula of annulment and on the ability to hear it (except in the case of the wife of a deaf-mute) prove that here we are not dealing with a condition. Perhaps this is connected to what I wrote in the previous point—that this is not an actual desire but a general willingness, not necessarily conscious, for there to be a possibility of annulment.

(*) This is Rashba’s wording (4:310): "…And in this matter, the ban of the community is more stringent than the oath of a married woman who vows while under her husband. Even though she explicitly said that she is not vowing subject to her husband’s will, nevertheless her husband annuls it for her. For the Torah placed her under the husband’s authority regarding every vow and every prohibitory oath that involves self-affliction or that is between him and her. And likewise under the authority of the father, whenever she is a maiden and under her father’s authority and has not yet married. Because most women think this way, and most of them vow with that in mind. But the community, since they act this way on their own initiative, it is enough for us to rely on their unspecified actions as based on this mental condition. But whenever their intentions are explicit, or in a situation where their intention is evident from their actions or from their wording, then the matter reverts to its original law, and there is no permission…"

As I wrote, Rashba’s words are very puzzling. If the Torah is what established this, then what does the majority of women have to do with it? Beyond that, it is clear that there can be a condition in vows, so regardless of the formula of annulment and the other halakhic requirements, and even if one holds that annulment is not a condition, still if it became clear to us that there is such a condition in the woman’s vow, one could annul it in any formula one wants.
In explaining Rashba’s view, one possibility is that the majority of women only determine the marriage contract—what the spouses agreed to there. When a woman consents to betrothal, she transfers to the husband certain powers over her, including annulment. The question is what exactly she transferred, and the intentions of most women testify to that. And when the majority changes, apparently the consent at the time of betrothal changes as well. Therefore, even if a woman explicitly says at the time of the vow that the husband should not annul it, that does not help, because already at the time of betrothal she had consented to this.
The practical difference is that if a woman says at the time of betrothal that her vows are not subject to the husband’s will—that would help, even if most women do not agree to this. And indeed this is like any condition in betrothal.
Still, I have always wondered why we never find a condition stipulated by the woman in betrothal (even "on condition that Father approves" is a condition from the husband’s side). True, the husband performs the act, but the woman’s consent is required, and she certainly should be able to consent conditionally. This requires further examination.

These things are a bit difficult for me, because a husband annuls his wife’s vows even when she was betrothed through her father, and even when she was betrothed on a rabbinic level through her mother and brothers.
And how can her mother and brothers transfer the rights of vow annulment to the husband if they do not possess them at all (for they themselves cannot annul her vows once she reaches the age when her vows are legally examined)? And further, how does the yavam annul her vows? Surely one cannot say that in levirate marriage she transfers the rights, since levirate marriage takes effect even when she did not consent to become the yavam’s wife at all (unless one says that already in her betrothal to her deceased husband she transferred the right of annulment to the yavam)?
And Rashba himself, in his novellae to Nedarim 73, wrote: "Rather, Rabbi Eliezer holds that since she is acquired by him, whether through betrothal or through levirate declaration, and her time has arrived, she directs her mind and makes her vow dependent on him, and this too is the reason for the verse, which says that in the case of a betrothed maiden, her father and husband annul her vows." This seemingly can be said only if this dependence is within the vow, not if it is some sort of agreement, because when she is still a minor she seemingly has no power to grant rights of annulment.
It must be, then, that even according to Rashba this is not a condition, but rather a general willingness on the part of most women, because of which the Torah granted the husband the ability to annul vows, not by force of a condition.

P.S.
Tosafot in Niddah 46 wrote (and Rashba in Niddah cited their words without disagreeing): "And if you say: if so, why is it written in the passage of vows that a husband annuls his wife’s vows even when she is not really his wife, as here, where the husband annuls because she vows subject to his will? One can answer that it is needed for the case where he said, ‘I uphold it for you,’ in which case he can no longer annul it. Alternatively, because the Merciful One said that a husband annuls, therefore she always vows subject to his will; and even though here there was only a rabbinic marriage and the Torah did not say that he may annul, nevertheless she thinks he has the power to annul as with other women, and so she vows subject to his will."
From their words it seems that this is a full-fledged condition. And both answers require examination:
1. The first answer is a bit difficult: what difference does it make that he said, ‘I uphold it for you’? At the end of the day the condition was not fulfilled. Also, if so, in a rabbinic marriage the husband’s upholding should not be effective—have we found such a distinction?
2. The second answer in Tosafot seems to mean that a woman vows subject to her husband’s will because the Torah placed the right of annulment in his hand—in other words, the cause and effect are reversed: not the women’s intention bringing about the right of annulment, but the right of annulment bringing about the women’s intention. But if so, this is difficult in light of the Talmud elsewhere, which says that we do not find one person forbidding and another permitting, and challenges this from a woman’s vows, and answers: every woman who makes a vow does so subject to her husband’s will. But that seemingly does not fit with Tosafot’s answer here, where the law of annulment precedes the woman’s intention.
And this requires examination, and may God enlighten our eyes.

Indeed, all this requires examination. But having the law precede her intention is puzzling. If in any case that is the law, why is her intention needed? These words were said only regarding rabbinic betrothal, where strictly speaking he has no right of annulment. But since she thinks he does, as in Torah-level betrothal, she transfers those rights to him.
But it seems more likely that the point is not that the law precedes the condition, but that the verse granting the husband the right of annulment reveals to us that a woman vows subject to his will. That is the reason for the verse. Still, the wording of Tosafot does not seem to suggest that this is what they meant, because otherwise they would not have needed to say regarding rabbinic marriage that she thinks he can annul it; rather, simply that she transfers him the right of annulment just as in a Torah-level marriage. Logically, what difference is there between this and that? So from their wording it really does sound as though the right precedes the condition, and that requires examination.

As for a minor, one can say that the condition comes from the husband: he betroths her on condition that the right to annul her vows be transferred to him. And I already noted that we do not find conditions stipulated by the woman in betrothal. Even when there are conditions for her benefit, it is the husband who stipulates them. And certainly when the condition is for his own benefit, as here (that he can annul her vows). She (or her father and mother) merely consents to it.

The difficulty from her father and mother and brothers is not difficult. If they have the power to give her over, then they also have the power to give her over on condition.

How can one discuss whether the husband cuts it off entirely or merely weakens it, if it is nothing more than a condition that the woman makes dependent on her husband’s will?

We already noted above that it is unlikely this is an ordinary condition; rather, it is a Torah law. Beyond that, why specifically on the day he hears it, and in a particular formula, and so on?
Actually, the question of whether he cuts it off entirely or merely weakens it does not strike me as difficult at all for the condition approach. She makes it dependent on the intention of both of them (the fiancé and the father), and so the question is exactly how her intention is divided between the two of them. What is the problem with that?

That is exactly what I came to prove, although from the Talmud in Gittin 83b it appears that the husband neither permits nor accomplishes anything through the annulment of his wife’s vow. I mean the passage in Nazir 22b, which discusses a woman who vowed to be a nazirite and became impure, and afterward the husband annulled it—whether by his annulment he uproots the naziriteship retroactively or only from then onward, in which case she still needs a sacrifice. But surely it is obvious that there is no possibility of stipulating a legal effect that will take effect and then later be uprooted from that point on by someone’s will or subject to some condition. Conditions can take effect only retroactively, or else not take effect at all.

I did not understand. I wrote that I do not see proof from the passage about cutting it off entirely, even though I agree with the conclusion you wanted to prove (as described earlier in this thread).
As for conditions that operate from now on, that is not precise. There are cases of from-now-on and retroactively, and the main proof is precisely from a vow, which is considered something that can be permitted, even though its permission uproots it retroactively. See Rabbi Shmuel’s lectures on Makkot, sec. 420.