Sure. Many medical conditions and procedures are biologically bound. Gender identity wouldn't override those practical considerations. It wouldn't be on an individual trans-man's gender identity to insist on a prostate examine and it be covered by health insurance for the same reason that no medically useless procedures need be covered by insurance. However, the same trans-man gets to pick his name and his pronouns for the same reason that you or I get to tell others we are men and use male pronouns. They are and have been generally bound to the person's self-identity. You're not going to know their sex anyway, so it cannot be bound there. Now, before you actually know them, you are likely either going to use 'they' or whatever you perceive their gender to be based on their gender presentation. If they don't have other input, it's your call. If they do, it's their call. If there are overriding medical reasons, then it is the call usually of the medical community. Gender expression, gender identity, biological sex.
Which sports, where, organized by who?
Yeah, another 'it's complicated' because it is. I don't expect the considerations of professional sports leagues to match high school sports, or to match pick up games on the corner, even for the 'same' sport. They don't have the same primary purposes, so what factors are the most important will change. Their organization change, and who gets to make the call change.
A professional sports league's primary purpose is entertainment. How it does this often takes into consideration its audience's sense of fair play, sportsmanship, competitiveness, tribal affiliation, etc (oh and advertiser input is almost as important as what the fans want). Barring legal considerations, the leagues have the call. It doesn't seem to be an existential crisis, and their considerations on what level to put the bounds at is beyond my pay grade. I just don't care enough about pro sportsball to keep looking into vague 'what about pro sports' arguments. Specific arguments may vary.
A college or high school sports league isn't supposed to have the primary goal of entertainment. It's supposed to teach values and offer physical challenges that build character. When I wrestled in high school, there wasn't a separate league for women because there were not enough women for it. They wrestled the men. It was the individual's call to wrestle or not, they didn't have to even make a choice on gender, and it worked fine. (Mostly, the cis woman gym coach in my school wanted to force any girl joining to take a physical test which was nonsense and in fact specifically barred by law in New York at the time.) Schools have state backing generally and who gets to choose there gets even more complicated even quicker. As a practical matter, trans gender athletes have been competing for a decade (except in Idaho) and none of the hypothetical concerns have come to pass.
Local sports generally don't have to let anyone play with them they don't want regardless of reason because they can, literally, pick up their ball and go home. The overwhelming games of sports played are informal and rely on mutual consent. They also tend to be co-ed. The point of trans gender people is moot there. It is always the call of the people involved and what they value. More and more are putting the value on inclusive and welcoming rather than the obsessive focus on validation through victory (which is always hilarious to me because they're all at best a big fish in small pond).
Most of the 'sports' I take part in are co-ed, and their primary goal is to build personal skills and meet other goals as set by individual participants. Rules change often to prevent 'training to the rules' and not the skills. Refusing an opponent based on gender identity, gender presentation, or biological sex would be laughable or disrespectful. But martial arts tend to be you competing against the you of yesterday, so only tangentially related to most 'sport'.
Regardless of whose call it is, it should be based on facts and real evidence and not unvalidated 'concerns'. WIAA have allowed trans gender athletes for ten years, the NCAA for nine. Trans women haven't been blowing away records or even taking top spots at a rate even equal to their participation. That's even without requiring two years of hormone treatment, which drastically reduces muscle building (skeletal structure is not effected). It hasn't effected the viability of cis women in those sports. Empiricism beats speculation. Many of the asserted harms don't make a ton of sense.
