Why only "possible"? Is there a side that says this is not a Sabbath-generated prohibition?

Two possibilities: 1. They did not decree a Sabbath-generated prohibition on an act that, according to the person doing it, is permitted. That is, a Sabbath-generated prohibition applies only if there is a transgression by the person in performing it—meaning, when he acted in a way that was not proper for him. 2. According to some halakhic decisors, a Sabbath-generated prohibition applies only when there is a change in the object itself.
I now thought that if the person acted unintentionally, others are permitted to benefit from it already on the Sabbath itself. When a person thinks it is permitted—and certainly when major halakhic decisors hold that way—that is no worse than acting unintentionally. Therefore, possibility 1 seems clearly correct.

According to this, would they also not have decreed a Sabbath-generated prohibition regarding an act done by a secular Jew who does not believe in the giving of the Torah, since according to his own view the act is permitted?

Also, even if there is no problem here from the standpoint of a Sabbath-generated prohibition, is there still a problem of directly causing someone to violate a prohibition?

And I also saw that you wrote elsewhere:
"Logic suggests that the act is done as my agent and for me, and that is like an act that I myself did intentionally, which is forbidden forever (not necessarily by the formal law of agency itself)."

And I also saw elsewhere that you wrote that we make an a fortiori argument from the rabbinic decree against telling a non-Jew to do something, and from there learn regarding telling a secular Jew, which is forbidden—so why shouldn't we learn from there by the same a fortiori argument also regarding telling a religious Jew in the example I gave above about the prayer book, that this is forbidden because of the decree against instructing someone?

A secular Jew is no worse than a non-Jew.
I think I wrote that about an act that I pay for. Here it is less likely to apply that.
Because here, according to the person doing it, this is not a transgression (even if I myself were the one doing it).

Why is this not "do not place a stumbling block"? Because it is not a case of "the two sides of the river"?

You wrote that not דווקא when payment is involved. See the responsa at this link:
https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%A9%D7%90%D7%9C%D7%94-%D7%91%D7%A0%D7%95%D7%92%D7%A2-%D7%9C%D7%A9%D7%99%D7%98%D7%AA%D7%9A-%D7%A2%D7%9C-%D7%94%D7%9B%D7%A9%D7%9C%D7%AA-%D7%97%D7%99%D7%9C%D7%95%D7%A0%D7%99-%D7%91%D7%A2%D7%91%D7%99/

As for what you said that a secular Jew is no worse than a non-Jew, I think there is a difference between the non-Jew of earlier times and the non-Jew of today. The non-Jew of earlier times was regarded by the Sages as someone who had not formed his view in a balanced way. But nowadays, as you said elsewhere, there are non-Jews and secular people who arrived at their views after study and investigation, and formed for themselves a worldview in which, in their view, the Holy One, blessed be He, was not revealed or does not exist at all.

As for this: "Because here, according to the person doing it, this is not a transgression (even if I myself were the one doing it)." According to the secular person's own view as well, there is no transgression even if I myself were the one doing it.

I didn't see whether you addressed the issue of directly causing someone to violate a prohibition.

Yehonatan, because according to the one being caused to stumble, this is not a transgression.

Oren, the difference between a secular Jew and a non-Jew is not relevant to the discussion. A non-Jew is not obligated in the laws of the Sabbath.
A secular Jew does not recognize the concept of transgression; a believing non-Jew does.

I understand.
Does the prohibition of directly causing someone to violate a prohibition apply to the example of carrying the prayer book on the Sabbath?
Because elsewhere you wrote this:
"As for telling a Jew, that is more problematic, since there is room to say that in direct causation the other person's act is considered your transgression (Rashi at the beginning of Matot), and if he is acting unintentionally or under duress then all the more so (Maimonides at the end of the laws of forbidden mixtures, and elsewhere)."

There is something akin to direct causation here. I'm not sure it is literally that.