I’ve had problems with the separation of sex and gender as concepts for years but I haven’t ever really put into words why I think it’s wrong. In a couple of sentences, you have managed to get to the core of my objection:
‘A case will be made that sex is distinct from gender — the former being purely biological in nature, the latter being entirely social. Upon making this claim, it will then be argued that, while trans women may indeed be women (because “woman” is a gender category), we nevertheless remain “biologically male” (a sex category). This line of reasoning is often accompanied by claims that women are oppressed because of their sex (not gender), and therefore feminism should be exclusively for “biological females” (thereby expunging trans women).’
And there it is, right there. At its essence, this is a TERF argument and trans allies have allowed ourselves to be taken in by it, giving currency to TERF views.