So many of your/our assumptions about
gender are inaccurate historically and grammatically as they pertain to science, culture and language.
No doubt it's unfixed definition is partly why it has become such an active battleground for agenda and bigotry. Are we talking words? are we talking primary sexual characteristics? are we talking observable behaviour? self identification? or do we not really care and simply use it as a way to promote wokeness/bigotry on the socials.
Biological sex is what it is.
Gender is either a social construct (i.e. 100% created by external influences) or it’s intrinsic. We’re born with it. It can’t be changed. This idea is central to the concept of people being transgender.
If gender really was just a social construct then that would imply you can’t be born trans. Obviously, that’s a very controversial claim.
I'm not quite sure if you are arguing a specific position on whether a distinction between sex and gender actually exists, or if you are just trying to catch people out. But:
As it pertains to behaviours (sexual preference, femininity, masculinity)
Born this way has always been a bad argument, at least philisophically speaking (controversial or not I've seen Gay/trans/me people all make this point). But that doesn't mean that it can't work as a well intentioned promotion of inclusivity.
Gender means different things to different people. Depending on your parameters it seems quite possible to view gender as partly arising from primary sexual characteristics and partly culturally learned behaviours.
Hell you can see it in dogs raised amongst other dogs of the opposite sex that will often cock/squat as learned behaviour independent of their biological instruction. Edit: This means nothing to the point about gender classification.
Gender as construction (partly or entirely) seems a solid position (see Beauvoir and many since for some of the better arguments and examples). And just because something has been constructed doesn't mean that it doesn't have a real affect psychologically or culturally on individuals or society as a whole. Surely we can try to understand and empathise with
feeling like a woman as an abstract concept, without having to get out the tape measures to see how they measures up to some womanly universal form.