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."
You know how women are. They HAVE to be righteously indignant about something because that gives them an excuse to virtue signal and attention whore.
This issue is retarded but at least it gives SOME women a pretext to virtue signal and attention whore against the gay commie agenda, rather than for it.