minority rights to be cast in any role in any movie

No, it's not really that. Rather, it is about the fact that people from minorities already have very few roles available to them which are not somewhat stereotypical and often have trouble being cast in roles that do not say anywhere the people playing them has to be white, and that therefore it is highly unfair if roles that were written as minorities are also given to white people. It's not about a privilege for minorities, but rather about white privilege still being highly entrenched in the movie industry.

As I am white and not even from the US, I am likely not the best person to explain this. There is lots of material available online on this issue, including things written by people who themselves belong to minorities and have first-hand experience. Just google "hollywood racism" or "hollywood whitewashing", and you will find many articles and interviews.

I'm sure it is not my place to say whether it would be OK for Koreans to play Japanese. But if something were cast like this, many people would likely see it as a sign of Hollywood insensitivity, as in "all Asian people being the same". And given Korean/Japanese history, I suspect that there might also be a negative reaction in these countries. As you mentioned "Memoirs of a Geisha": There was indeed some negative reaction to the casting; on that, too, material is easy to google.