2 things: I've asked this before, and I apologize if you answered before and I just missed it, but what is Dr. Mrs.'s opinion? Given that she has a doctorate, she is clearly a competitive person. I would be interested in her opinions, if nothing else to see where they differ from the women in my own life.
On sports, I know she values the human right over a supposed sanctity of the competition. Her attitude towards all of our institutions and activities is "if tolerance and egalitarianism mess with you, the worse for you." So sports is a bad example.
I will pose a thought experiment to her to try to short out her wires. Posit STEM scholarships for women. How do you feel about trans women receiving them? Should there be some mechanism for verifying they are women? By my prior sweeping generalization, there should not. The rightwing bugbear Mr. T should be able to walk in, say "I identify as a woman," and qualify.
That might be a problem for her.
Does it change your mind at all if we get away from individual events like swimming and track and into contact sports? A previous poster mentioned how playing boys hockey got more dangerous for them as they all got older. Girls wrestling is becoming bigger in many states at the high school level, and women's boxing and MMA fights are already big professionally. Are you still ok with a trans woman competing in those sports with no restrictions or requirements to have been on hormone blockers for a set amount of time whatsoever? Participation still trumps all?
On the individual high contact sports like wrestling it's just like going up against a very strong cis woman.
On team sports my initial reaction is a pause. That's interesting. I'm certainly wrong and it's some latent bullsh-t in my beliefs, but I don't know where it comes from. That's a good experiment. So is the justice one I'm going to ask my wife above, although for me personally that's not a problem because, for me, just because some might take advantage is no justification for denying a person their rights.
But these were both excellent questions.
This may sound strange: I don't think we should be guided by principle, here. I know that seems weird, especially to rationalists like you and I. What I mean is, this is an area in which intellectual reasons to justify harming people are still fairly robust. We can't do that with slavery anymore, and intelligent people can't do it about reproductive rights, but trans rights are "new" to us and so there is still a lot of gross rationalization washing around in our heads that justifies making them contingent on other priorities.
Recognizing that, I'm going to go with "first do no harm." If it seems to violate a principle of fairness not to hurt someone, then that principle of fairness has not caught up to reality yet. I know that seems dicey, but it's the closest I can come to describing the experience of realizing some of my intellectual conclusions are nonetheless wrong.