First, trans women are women.

Second, any artificial environment will select for some subpopulation. The more restricted the criteria, the more one can expect to favour particular groups of people.

What do I mean by artificial environment? Any environment which does not appear in nature without deliberate human action. A hospital is an artificial environment which selects for people who are sick and healthcare professionals. This is expected, the purpose of the hospital from the onset is to provide healthcare and therefore the inhabitants of the space reflect that. Does that mean one may not see, say, a clown? No, only the likelihood would be low, that is, one would not expect to walk in to a hospital and see clowns making a majority of the people. It certainly seems rather unlikely

Sports are artificially environments which select for the kind of athlete well-suited towards the particularities of the sport. This means minute average physiological differences which conventionally do not have much survival value (as in, survival in a natural environment, the kind humans inhabited exclusively prior).

If we accept that besides the ability to give birth, there is no meaningful difference between any genders or sexes, as survival is not contingent on 'muscular strength' or 'flexibility' or 'height' then we can see that in the patriarchy which obviously favors men in specific contexts, one would expect men to succeed. This is the case, and then the considered 'superiority' is then valuable.

The patriarchy is relevant because it squashes all similarities between individuals (of which there are more than differences) and highlighting the similarities necessarily decreases the 'perceived value' of the differences, again such as average height, muscle mass, etc. Note, there are differences, they are always present, the issue is with 'perceived value', not whether differences exist.

In short since survival and competition have largely been eliminated in contemporary human environments, it is reintroduced in a way to give meaning or satisfaction to the involved parties, a simulacra. This is confused with a genuine natural environment, which does not place importance on whether one is cisgender or transgender.

To give an example, it is uncontroversial to state that if I picked any two random people, one who is fit instead of out of shape, the fit person would likely be the winner in a 100m race. Ok, let us use what transphobes and bigots use as reasoning. We will say that the fit person is a trans woman and the out of shape person is cisgender. This distinction becomes moot when one thinks about management, that is does it matter what the gender–or whether a person is cis or trans–is for the purposes of winning? On the whole, and not only involving the observable competitors?

No, it does not. Once we accept that the argument is contingent on a narrow set of criteria which tend towards the result of transphobia then it can be accepted that any broadening of the environment, describing how it actually is, removes any 'perceived' differences in ability. A woman coach is equally capable as a man coach, as is a trans woman compared to a cis woman, and as is an intersex person to a non-intersex person, etc.

If coaching had a similar tendency towards one gender, somehow, one could move up the ladder. Is there a meaningful difference between the owner of a sports team, between a cis woman and a trans woman? No, there isn't, the differences mentioned above only matter in the narrowest sense. Bigots and transphobes work with the existing artificial competitive environments, sports, which do not have much meaning or value for most people in most situations. Neither did such artificial environments have a meaningful impact on survival in early human history. Obviously sports matter to those involved very much, and that is a subset of the population.

If having higher muscle mass, height, and arm span mattered; why did Homo sapiens outlive and outcompete Homo neanderthal, the larger and stronger hominin? The simple answer and one which can be well reasoned and understood with empirical evidence (empirical, rather than idealistic or fantastical) is that the emphasis on increased strength and height do not matter very much. Perhaps 'intelligence' or 'adaptability' or 'endurance' matter more. Or, perhaps strength really does matter more and Homo neanderthal went extinct for a different reason. The point is it cannot be the sole determinant or as important as conventionally thought, which is what is typically argued for. Human bipedalism, society, language, culture, 'intelligence' etc. are all apparent significant factors to our species survival and thriving. We are no longer in the same natural environment, thus the utility and value of characteristics have changed.

Focus on specific characteristics is good for thought experiments and to determine the limits and to break boundaries, and it is not nor will ever be the most important factor.