Sorry, I misunderstood what you were asking, but I have also already answered this question. I am fine with getting rid of the segregated leagues in pool AND I'm also fine with allowing trans-women participate in the women's league. Either way is good.
As I have also pointed out (but got shouted down because I used the taboo "feminism" word), there is a historic gender role difference in playing pool. I think I also mentioned that it's similar in chess, but that's a different topic. The segregated leagues have an advantage at letting women feel comfortable playing in a sport that is historically dominated by men, without the harassment and other social pressures. (I'll refrain from using the also taboo "safe space".)
Thanks, I do consider this to be a real response to the question. When it comes to pool, you don't think males
do have an advantage over females, and so don't think that transgendered males (trans women) should be prevented from competing with women, not because they are different from other males in not having an advantage, but because there is no male advantage in general.*
If that were the case, I'd be sympathetic to that view (I think I said so in response to your first post in this thread on the subject of pool), though I have reservations which are actually associated with your second paragraph above: whatever the reason that women are underrepresented in pool, and which thus justifies segregation based on sex, I suspect it applies in general. But I am somewhat
less concerned about socio-cultural factors that affect representation that biological ones.
That said, I
do think there's reason to think that the differences in representation are biological. The evidence that you presented, showing that of the top 100 highest rated players only one (in 59th place) is a woman suggests that
something is going on. It certainly could be socio-cultural factors. But there are also good reasons to expect males to perform better in physical activities in general, and Ziggurat earlier gave what I consider convincing arguments for physiological differences that apply to pool specifically.
None of this leads me to certainty, but based on the evidence we have, the best theory seems to me to be that there are meaningful physiological differences leading to the differences in outcomes between the sexes in pool, and that these differences apply with respect to trans women as well as other males. Given that, sex based segregation (as opposed to gender based segregation) seems reasonable.
If evidence were presented that the differences in outcome
are not due to physiological differences, I'd update my view, and think there was less justification for sex based segregation, but I wouldn't think there was
no justification for it, as explained above.
*EDIT: I paraphrase your viewpoint in part to check if my interpretation is correct, obviously if that paragraph
isn't what you think, then please correct my interpretation.