Except 'legitimate' sports treat the events equally. The last two Olympics (Winter and Summer) no longer have male-only sports. There are only certain divisions within a sport that have male-only or female-only contests. But, due to the drastically different playstyle and nature of each individual game, they shouldn't be considered 'divisions' but completely separate events - and thus support, at a minimum, male and female divisions for each.
Additionally, non-physical 'sports' (Chess, Racing, and Gaming) have traditionally allowed male vs female competitions. In most cases, they may have female-specific leagues but those are only to encourage female participation, not as a means of segregation (females are allowed to participate in the main event).
Plus, America has Title IX, which prohibits collegiate sports (and education) from discriminating against female athletes by not having (or at least allowing) equal support of female leagues (it does a ton of other things, but that's the relevant point). And most 'legitimate' sports have some form of collegiate level support.
So, TLDR: The Olympics don't support this, there's examples of this decision not being applicable to other similar legitimate sports, and it would be illegal if they attempted this in the States at a collegiate level (where most legitimate sports have support).
Basically, this is the exact opposite of what they should be doing to push for legitimacy as a sport.