Answer

On the face of it, annulment of vows is only a clarification that there really is an opening, but it is not needed for the annulment itself. But in practice, the accepted view is that Torah law requires that the vow is not permitted without an act of annulment; that is, the sage’s action has constitutive status. You are asking why that is so.
The question is a good one, and Maharit already addressed it in section 19, bringing in the name of his father (the Mabit) that the need for annulment is only rabbinic. See there for the full discussion. But it seems to me that this can be explained more broadly. In my view, the “opening” discussed in the laws of annulment of vows is just that—an opening, meaning not an absolute consideration. If it is completely clear that this vow was never included in his intention from the outset, then perhaps it is void on its own even without annulment. An opening and annulment are required when the opening is not unequivocal. And there are opinions that regret is also required, not only an opening. There is further support from the view that it “cuts it off,” meaning that the vow takes effect and is then uprooted, rather than our discovering that it never existed in the first place. That is also how Rabbi Shimon Shkop understands the Talmudic passage that rules that a vow is something that has a means of becoming permitted in tractate Nedarim 52; from here we see that the vow is not retroactively nullified, but rather it is “cut off”—from now on, with retroactive effect.
As for betrothal, in a case where there is a clear consideration that “she would not have accepted betrothal under such circumstances,” the betrothal is indeed void (although the halakhic decisors tend to avoid applying this rule, because of the presumption that “it is better to dwell as two than to dwell alone”). But here too, an opening is not enough (and certainly not mere regret); rather, there must be an absolute and clear consideration.