Incidentally, there were a good many Asian faces in Firefly - quite a few of the extras in crowd scenes were Asian. That doesn't address the lack of speaking roles, but the need to see Asian faces in order to support the premise of a Chinese/American cultural dominance was at least attempted.
Why, given their supposed dominance, not one Asian would happen at random to be part of this fairly randomly-assembled group of travelers always seemed to me to be a question that begged an explanation.
I suspect Whedon would have one, if pressed. It might not be one that he was really comfortable with disclosing. He may have assumed pretty rigid social segregation for some reason.
Certainly his casting of Mal as an equivalent to a American Confederate soldier was something that he always felt was pretty dicey but still worth doing.