Regarding my point B: If somebody threatens to do something bad (check genitals) to check for a condition X and then somebody else suggests doing something else (take their blood for a blood test) to check for a condition X.

This does not make the second suggestion a good thing, it is merely a lesser evil. One should first check if the condition X needs to be known. In the case of bathrooms I'd argue that no it does not. In the case sports generally they would need to disclose it but I would like to note that it probably doesn't need to be publically known.

I just thought you might like to hear that the International Olympic Committee officially allowed trans athletes since 2004, while the International Association of Athletics Federations first allowed trans people who transitioned early enough. Nowadays both of their policies appear to be mildly more inclusive than they were originally.

At least according to the following sources after one or two years on hormone therapy when accounting for the weight difference (which I believe most sports already do and did previously as well by setting weight classes) there is no performance difference between transgender athletes and cisgender athletes. (One exception being that trans men completed more sit ups in one minute)

Sources: Effect of gender affirming hormones on athletic performance in transwomen and transmen: implications for sporting organisations and legislators

Effects of gender affirming hormone therapy on exercise performance in transgender athletes

Correction:Strength, power and aerobic capacity of transgender athletes: a cross-sectional study

I haven't looked specifically for it but I did not find a source claiming otherwise when trying to research trans athletes on hrt.