Should Trans Women be Allowed to play in Women's Sports?

Yes. Reasoning? Trans women are women and women's sports should be open to all women.

"But what about their biological advantage of being AMABs?" Some humans are physically more capable than others! This is not only normal, but professional sports are largely about those more physical capable than their peers. However some might think the performance gap between transfems who hasn't gone on HRT yet, and all other women (including transfems who have gone on HRT) is too large and would diminish the participation of cis women, and trans women on HRT. That is a valid criticism! I find that transfems needing to be on HRT to compete is an entirely valid requirement to narrow if not downright eliminate the gap in average performance. (Studies have shown HRT is very effective at this performance reduction). That does come with a caveat of those transfems wanting to compete must be given access to long term HRT, if you deny them this, then the argument to regulate them is moot, because its removing the possibility of women competing with other women. Of course the performance difference between cis men and cis women in many sports is lower than you'd think. It's really not as big of a deal as it's made out to be.

"If it doesn't matter if people out perform each other, then why have a women's league at all?" Good question! Now we're getting to the heart of the matter, the reason women's leagues exist isn't because men are inherently more physically more capable (most professional women's athletes would stand head and shoulders above most men in their respective fields), it is instead because of one crucial fact, misogyny. In open leagues of any sport and/or non-physical competition women are frequently harassed, belittled, and subjected to increased hostility from others, and this removes a large portion of them from competing that otherwise would because they simply do not wish to put up with it.

"But what about trans women harassing cis women?" It's true that women sometimes harass other women, but you'll note that trans women are more likely to be the subjects of harassment due to wide spread transphobia and more specifically the unique intersection of transphobic oppression and misogynistic oppression known as Transmisogyny. If a women's sports league is about participating in sports in an environment with less misogyny (women can still be misogynistic, and worse a lot of women's sports are administrated by men which leads to systematic misogyny in a sports category meant to protect women from misogyny) then excluding trans women from women's sports would have the effect of forcing trans women to compete with men and thus subjecting a group of women to both transphobia and misogyny if they wish to compete (not even mentioning how doing so would be systematic oppression itself). Something that as discussed above will cause very few to actually put up with the systemic abuse and participate, thus effectively banning them from sports not in name but in function.

"But trans women aren't women." That is factually incorrect and requires willful ignorance to believe, I can't logic you out of a position you didn't logic your way into so I won't even try.

"Gender might be real but misogyny is sex based oppression, not gender based oppression." That's a TERF talking point, and also factually incorrect, and I'll at least link to a post about it. Though again it has no basis in how misogyny is actually applied in real life.

"But what if they're cis men pretending to be trans women to out compete women?" Well if we institute the hormone balance testing for trans women (along with a minimum period, and of course actual access to trans health care in the first place) then it doesn't matter! Mostly because no cis man would subject themselves to years of hormonal gender dysphoria, misogyny, transphobia, and transmisogyny to beat women at sports. It's a null argument because it's not actually happening and a lot of systematic bigotry (the same bigotry that creates the need for a women's category to begin with) would actively prevent it. If one person gets through, honestly that'd be the most impressive stunt ever and would be beautiful to behold… but they'd likely be the most dense transfem egg in the universe rather than an actual cis man.

"The logic of women's leagues being about being free from oppression seems to check out but what about making new leagues for transgender individuals and other marginalized groups instead of putting them into women's sports?" This is an actually interesting question, thank you! Separate leagues to support marginalized athletes isn't exclusive to misogyny! There have been many leagues for racial minorities in the past. These fell out of favor for reasons I haven't actually researched! That being said, transgender individuals make up such a small portion of the population that a separate league to shelter them from transphobia would have next to no funding, visibility, nor international participation... if that wasn't the case then it'd be really cool! However, the first point still stands, trans women are women, and they should have access to women's leagues. If there existed a separate league for racial minorities to protect them from discrimination, it wouldn't make sense to ban racial minorities from mainstream sports, same with gender & sexuality minorities, if such a league existed using it to bar transgender individuals from mainstream sports participation, it would be a very extreme form of transphobic discrimination. If you'd support such overt discrimination, fuck you!