But that’s not how it’s defined. People have been using that word in every language humans ever invented for thousands of years to mean biological female. If you want to argue that there is something else that isn’t biological sex and you want to invent a word for it, go nuts, but “woman” is already defined. Words can and do change definitions over time, of course. If it’s your contention that the definition by consensus has already changed, say so, but there are billions of people on this earth who haven’t got the message, which seems odd for something determined by consensus of the people who use language.
Putting that aside, since sports are about physicality and accomplishing things in the real world, it makes no sense to base them on “identity” - something that cannot be detected or defined by anyone but the self identifier - rather they should be based on physical aspects of reality.