I'm not trying to define away biology here. Although "sex" is surprisingly hard to nail down.
Rather, I'm arguing the underlying motivation for creating these categories was and is a sociological one. Why carve out womens sports, as opposed to short peoples sports, low testosterone sports (or other categories which would be similarly disadvantaged)?
The only reason people pay attention to sex here is sociological, i.e. because of gender. This implies that the admissions criteria do not automatically have to follow these strict biological lines -- and I see little reason to enforce them this strictly now. Why exclude trans people and why make yourself a headache trying to classify e.g. intersex people?
More of an aside: a society which fully accepted trans women as women would think looking at the biological markers you're looking at is complete nonsense. Suggesting trans women should be banned would be as ludicrous as suggesting all women with a specific gene which might increase your chances of winning should be banned.
We carved out women’s sports because otherwise there would be no biological women in competitive sports, and that was considered to be a significant enough exclusion of half the human population as to warrant such direct intercession.
Whether or not a similar case can be made for other categories does not have bearing on the case for sex categorization. Such claims can and should stand on their own merits.