Is your argument actually that women don't generally compete with men in sports because the sports don't want to embarrass the male athletes if they lose? If so, I suspect this is a bad faith argument, but if not, you can simply do a little searching to find that there is often quite a bit of difference between the performance of top tier male athletes and top tier female athletes. For example, no woman has ever run a 4 minute mile in competition and more than 2,000 men have and even about 30 high school boys have. I am sure you can find other examples.
Why did Lia Thomas go from being nowhere near winning in the male division to getting fifth in the women's?
When competitive fairness is raised as an argument for segregation, it is pretextual
If sports were not sex-segregated, most events would never be won by a woman. How is that a pretext?
The moralizing parts of the conclusion of this article rejects it's own evidence. There are multiple studies cited by the article where the population average of the trans women group statistically significantly exceeds that of the cis women group. The article concludes:
"The exclusion of trans individuals also insults the skill and athleticism of both cis and trans athletes. While sex differences do develop following puberty, many of the sex differences are reduced, if not erased, over time by gender affirming hormone therapy. Finally, if it is found that trans individuals have advantages in certain athletic events or sports; in those cases, there will still be a question of whether this should be considered unfair, or accepted as another instance of naturally occurring variability seen in athletes already participating in these events."
Does it really insult the skill and athleticism of cis and trans athletes to exclude trans women from women's sports? I don't think it does, but the article could not help but claim that it does. Often in debates such as this one, there are multiple levels of sophistry that annoy me. Such as the sequence 1. there is no evidence that trans women have an advantage over cis women in sports (false. there is evidence) 2. if you believe that there is any evidence, you must be a bigot (well, obviously untrue, there is evidence).
Women's sports leagues often emerge due to the easy bifurcation of the population into two groups- the easiest fault line to judge this as is 1 group with the athletic benefits of natural testosterone, and 1 group without the athletic benefits of natural testosterone. People are free to make whatever sports leagues they want, and with freedom of association they can make whatever rules they want. I will just find it completely unsurprising that the women's divisions will be relatively "closed" and the men's (or more acurately the "open") divisions will include any person that has produced testosterone naturally or become a trans man (or most things in between). It's the easiest bifurcation that reduces questions of fairness. Weight classes in wrestling fall into a similar manner of thinking for me; even if it could be argued that the guy that barely couldn't make it into a lower weight class should be fighting within that class, you have to draw the line somewhere.