[…] an overwhelming majority of men have a bundle of cognitive traits no woman has and an overwhelming majority of women have a bundle of cognitive traits no man has.
Isn’t it just a matter of degree, though? I’d rather put it this way: although there are genetic differences with respect to both the cognitive and personality traits of men and women, there’s significant overlap between the two (overlap that will differ, naturally, depending on the trait in question). I’d also say that the differences in cognitive functions are far less pronounced than those in personality and behavioral predispositions, such as mating strategies, preferring people over things, etc.
It seems likely that the same-sex attracted would disproportionately turn up in the cognitive-traits-bundles overlap group. This would imply that gay men would often display cognitive patterns that we more commonly associate with women and that gay women would often display cognitive patterns that we more commonly associate with men. (This in addition to having the attraction focus that we associate with the opposite sex.) Which is, of course, exactly what we do observe.
In some traits that’s not the case. See here, for example (relevant extract in the image below).
You may have already read them, given your interests, but if not, these two books would provide valuable insights into some of the topics you’re covering:
David Buss’s “The Evolution of Desire: Strategies of Human Mating”