Many people here - the majority, I think - support the rights of men who want to do so to wear dresses and makeup and behave in what might be called a traditional feminine way.
But men who do, do so as men and in spite of being male, rather than as an expression of being women, which they are not.
Whilst women & men certainly share behavioural expressions, there's no denying some are more common to one sex than the other hence the GC desire to exclude men from women's private spaces because of an increased propensity to violence.
When trans women say they are women they are saying in essence their inclinations/behaviours are more associated to the more common/typical behaviours women than men & as such identify with them more rather than saying certain behaviours are sex exclusive as in stereotypical.
You can't have it both ways.
I think you've misunderstood. Trans people co-opt, utilize and leverage stereotypes in order to claim - for the purpose of claiming - they are the opposite sex. That's a rather different matter. Look at me - I'm wearing a frock therefore I'm a woman. No, you're not. You're a man in a frock.
So on one hand you affirm the sexes share behavioural expressions but on the other any shared expressions by trans people are 'fake'/copied & purely motivated by a desire to claim they are the opposite sex rather than an organic inclination that's commonly aligned with the opposite sex? That's incoherent.
This is what I mean about the GC blank slate blind spot. It's anti science in that it is not at all scientifically controversial that gendered expressions are influenced by genes & hormones. The idea that people don't have any intrinsic capacity for gendered coded expression is false as is the idea that common gendered behaviours don't exist as a result of this that has implications on how individuals might classify/associate themselves more with.