I don't disagree with any of that, but I would say that "trans women are women" is about gender, not sex.

It's unclear whether sports are segregated on gender or sex lines, but the historical record leans more toward gender segregation than sex segregation. I honestly don't care enough about any sport to really know the correct way to organize a league, but I do want trans persons to be able to participate in public life, including sport, as safely as anyone else. I'd mostly be in favor of no division based on sex or gender, but people that are more interested in sport than I tell me that's bad and unsafe.

I think that is most often true. However, there are risks to being outed if you are stealth and those risks could be more significant than the risks of being treated as the gender you present than as the gender you were assigned an birth.

The treatment of a broken bone is largely independent of the mix of sex hormones, presence or absence of sex-differentiated organs, or gender presentation now or at the moment of birth, but can still involve an ambulance ride and a hospital visit. I'd hope our medical system would be trans-safe enough that disclosure would be a strict positive, but I don't know that's the case, and I would defer to the patient's judgement about whether to disclose or not for whatever reasons (including but not limited to safety and comfort).