I think it's dumb to put that much importance on sports. But I've always been in favor of letting people associate with whoever they want, and not associate with whoever they'd rather not.

Yes, this means I would respect "whites only" clubs, as well as "blacks only" clubs, and "female only" teams, as well as teams that let anyone play. I think the smartest solution is to simply divide into leagues of similar ability-- keeping sex or "gender" out of it entirely.

Every human alive has equal and identical rights. Those rights don't include imposing yourself on others. You can cosplay as the other sex, but if someone notices you aren't what you claim to be, you need to move along and find a different team (or whatever) to play against.
(Just in case it isn't clear, I don't think sports is any of government's business.)