> None of them were claiming they were the only people who thought "rationally"; rather, they were mostly claiming that only rational thought could reach truth, and empirical evidence need not factor into it.

This is not my understanding. Rather, the disagreement relates to the primacy and necessity of rational thought in some places. Check out https://plato.stanford.edu/entries/rationalism-empiricism/ and you can find two examples of possible interpretations of each position. To be a rationalist does not mean to say that empirical evidence need not factor into knowledge, but just that some is unobtainable via experience, or that gained via rational thought is superior.

There's no doubt multiple views people have held that fall under each umbrella, but I just wanted to highlight that I think the rationalist does allow for some sources of knowledge that are obtainable only via experience. And I can't see how they could deny that experience is required for some knowledge.

(I may have gotten some subtleties wrong, it's a long time since I looked into this)