The arguments advanced here prove too much. Why have women's divisions at all? If a trans woman's physical superiority is just "biological good fortune," so are the advantages of cis males vis a vis cis women. Let every human compete, and let the cookie crumble. Surely, if winning isn't the point, boys and girls should "compete" together. They would all get the exercise, right?

Because the segregation of athletic competition by sex is about physical advantages, only a person's assigned sex matters, as that is the sex that (statistically) affects physical attributes. A person's "real" sex, or "chosen" sex has no bearing on their physical attributes and so has no place in deciding against whom that person competes in sex-segregated sports. When trans woman are banned from competing with their assigned sex, they can make a fuss. Until then, barring them from competing as women is simple common sense.