I largely agree, but I will add that we generally do allow people to self-identify their race. Lots of college and job applications that ask for your race will use this type of self-identifying terminology. The problem is when people are perceived to be lying about their identity as a means of getting some sort of advantage. Rachel Dolezal for example was caught lying about her parentage. Trans people generally don't lie to cover their transness (except maybe socially to avoid the attention their transness would garner).

This lying to get an advantage can potentially apply to trans people in sports too. People are always claiming that near elite male athletes who can't win in men's competitions will just say they are women to face easier opponents, but we haven't really seen any evidence of that occurring. This is primarily because the obstacles to check a box on a college application are so drastically different than the obstacles in front of someone who wants to publicly and officially change their gender. Considering that, I think the best approach is to not waste too much energy on "fixing" this problem until it is actually shown to be a problem.