Studies have shown that rich females are less willing to "date down" than males are.

Care to point to these studies?

Men aren't sexually attracted to money at remotely the level women are. Surely you're aware of this? If not, there are plenty of studies pointing in this direction, plus an evolutionary explanation, plus mountains of anecdotal data that anyone with exposure to other people has.

men don't care how wealthy their sex partners are

I’d even argue it goes in the other direction to a degree. I’ve witnessed many relationships where the woman makes more money and it creates tension. Obviously, not in all cases but It does happen.

Indeed, and even if the woman does not care for it, the man might feel lesser of a man because of it.

On the other hand, if a man is happy in this situation, he's being called out as Gigolo :)

Palimony is a thing, but what makes these males stick around?

Ask some men you know.