rich women

This, so much! There was a story floating around about some pool players throwing a fit about trans women, because they had been "male socialized", as if being rich enough to own a pool table and having time to play isn't a million times more important.

Yes a great point. Unfortunately transphobes and bigots really couldn't care less. I think one can convince those who have not convinced themselves completely, or who have only passing familiarity with the 'controversy' by explaining that if there is a meaningful advantage, the focus on the advantage by the specific characteristic of whether a competitor is trans or cis, is one factor and with available evidence not a major factor.

I say it is a factor because I imagine a person who has successfully transitioned to the extent they wish barring any barriers (like availability of medication, whether gender-affirming surgery is available, etc.) and thus they have been 'treated' or rather have at their disposal the resources they need.

This would be in contrast to a trans person who was not given the resources they need, was unable to transition say due to lack of safety in their environment or medical resources, and thus have additional barriers to face. If a suburban amerikan woman of an ethnicity which is the plurality of her environment was transgender, I would expect them to have a considerable advantage over a disenfranchised minority ethnicity in a rural region which does not have adequate medical resources or a suitable community to provide needed support.

The disadvantage is not intrinsic to whether a person is trans or cis, it is due to the material resources available to each party, which is known to be palpable and insufficient for many if not most trans individuals.