I heard President Trump finally gave in on this issue (depressing), but even his speech on it didn't give actual reasons.

"They'll be outcompeted" isn't an argument when the very institution you're worried about provides no value to society, sucking up cash to produce famous lesbians who hate men. The world definitely needs more of those, said nobody ever.

In my opinion, the argument is framed entirely wrongly. It shouldn't be "save women or you side with trannies", it should be "Repeal Title IX and kill women's sports off for good."