You make an interesting point, I think it's possible for someone to play another of a different ethnic background, however I'm pretty sure that we also have a lot more examples of English characters being played by English actors, French by French, etc. in Western cinema (and) Star Trek.
Just to use your sig, Indiana Jones ("American") played by an American, Harrison Ford. If Indiana Jones was played by a Mexican actor, maybe that could work too. I'm of the opinion that Picard was rather English in his sensibilities, most notably affectations for Shakespeare; also Pierce Brosnan's 007 was more generically secret agent than British, after all we could go to Scotsman Sean Connery and serving England, etc.
I suppose Harry Kim was more American than anything else, although promotions and all he didn't receive much character development actually. It comes down to the portrayal of minorities on TV, do they actually bother to have the actual minorities being represented cast or not? Should they? I think that it makes sense for that to be a consideration, out of many.
I should mention also that we did get Hikaru Sulu (Japanese) played by George Takei (Japanese American). And Garrett Wang was the first main Chinese cast member on Star Trek, it's just that his character wasn't. Hence the topic of this thread.