It is not and has never been rooted in any sort of sociological concept of gender as an independent category from one’s sex.
It is infuriating how successful the "facts don't care about your feelings" crowd has been at pushing discriminatory legislation through in the last few years based largely on feelings rather than facts.
That the criteria for admission are altered now to exclude some of them is motivated by anti-trans politics. Usually such rule changes are made when it becomes obvious that the old rules cause outcomes which go against the spirit of the sport. You cannot argue this here in good faith. There are not a lot of trans women competing and none have even won anything afaik.
People caring about this issue in sports now and changing the objective admission criteria to exclude them is a political phenomenon more than anything else.
This issue genuinely confuses me — and I don’t seem alone in that. Re-defining words does not redefine categories or change the underlying motivation for creating categories in the first place.
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.
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.