I assume that throughout the generations there were halakhic disputes about things that are indispensable for conversion… If so, according to what you write, we should have been fragmented and divided, yet it seems that in all generations those who disagreed still accepted even the converts of their halakhic opponents. Am I mistaken?

Yes, you are mistaken. Nobody accepted a conversion without acceptance of the commandments or without immersion. The disputes were about the scope of the commandments, the level of verification, conversion for the sake of marriage, and the like.
Just as they did not accept divorces from a civil court, as the rabbis of France wanted to institute in the 19th-20th century. Disputes are legitimate within a minimal halakhic framework.

Rabbi, if we take one step back, what are the criteria for accepting a religious court as a legitimate court for purposes of conversion? Suppose I do not accept their ruling regarding the scope of the commandments, the level of verification they actually did with the convert, etc.? That is, suppose they do agree that acceptance of the commandments is required in principle, but there is a dispute about the details.

A recognized religious court with Torah scholars as judges is presumed legitimate. But that does not give approval to everything they do. If they go beyond the reasonable bounds of Jewish law, their actions should not be recognized. Ordination to judge is not a license to kill, 007-style. Factual inquiries or choosing among existing views—that is the mandate of a religious court, but not decisions that contradict Jewish law. Nobody suggests recognizing Reform conversions, because that is certainly outside the bounds of the procedure (and of course the “judges” are also disqualified).

Sorry, I did not understand where the line is drawn. If one of the medieval authorities held that immersion before a religious court is indispensable and another held that it is not, because they disagree about understanding the Talmud, would one not accept the converts of the other court?
And if he would, then how is that different from the need for acceptance of the commandments? Why is that view not legitimate? I think those rabbis sincerely think and are prepared to prove that acceptance of the commandments is not indispensable… so why here is it outside the reasonable bounds?
Everyone has his own reasonable bounds. As long as we both agree on the principles—that the Talmud is binding and we are only arguing about interpretation—then why should we not accept their converts?
I think it is a mistake to think that way, but even regarding immersion before a religious court there were medieval authorities who thought it is indispensable, and whoever thinks otherwise is mistaken; so why there did they accept their converts?

In many disputes I think religious courts or rabbis adopt approaches that are contrary to Jewish law, as I understand it, and nevertheless they are still within the framework, and if I go to the place of a certain rabbi I will act in accordance with his understanding even though in my opinion it is a mistake—I am speaking in principle. In your opinion, if it is clear to me that that rabbi is mistaken in his ruling, do I not need to listen to him? In such a situation, does the prohibition of “do not form separate factions” no longer apply?
Why is choosing among existing views legitimate? Is it because they exist that they are not mistakes? And how is a view I invented today different from a view that a medieval authority invented a thousand years ago? A view is a view and it should be examined on its merits, not by whether it already existed or not.
If they think there is room for such a view within the halakhic framework, then what does it matter if I think not? This exists in almost every halakhic topic, where we find positions that can easily be rejected and shown to be baseless, and yet we still treat them as legitimate in order to sanctify autonomous halakhic ruling.

The line is drawn where I decide it is drawn. If you expect me to recognize an act of a religious court, I am the one who determines whether I recognize it or not. Every person has two boundaries in this context: the boundary of error and the boundary of legitimacy. There are legitimate errors and illegitimate errors.