I have seen it argued by pro-trans activists that asking people to state their pronouns when everyone is going round introducing themselves is itself problematic, precisely because there are people who identify as trans but don't want to have to out themselves. So if, say, a man who identifies (but only internally for now) as a woman is asked to state their pronouns, they either have to say "she/her" and invite incredulity, or "he/him" and deal with the internal stress. (I'm not endorsing this, merely reporting it, although it seems to make sense once you accept the premise.) Or they can say "I'd rather not say", which is itself outing.
(There are also some pro-trans people who reject the idea of being non-binary. At least if you are born a man and want to identify as a woman, you are accepting that men and women exist.)
Part of the stupidity of <waves hands> "all this" is that nobody on the happy-clappy side has sat down and thought about the logical consequences of their lines of argument, as shown above. Another is that OP's position of not recognising the concept of gender identity (which, unlike biological sex, is entirely socially constructed) is equally valid. Indeed, the claims of the colleague mentioned in the OP reinforce that — if they are allowed to not have a gender ("agender") then surely everyone else can. Or is "not having a gender" also a gender identity? 🤔
The whole thing is philosophical bobbins, but I suspect that if you took this kind of argument to HR, it would end with them saying (after checking that nobody can overhear) "look, @wherearethemarsbars, can't you just play along, we have to be nice to these people, otherwise they will call us bigots and the company will be getting boycotted". OP's position is probably shared by over half of her colleagues, they just don't have the courage to say so (often for very good reason, I should add — we can't all be Tank Man or Rosa Parks).
One bright spot is that after the SC ruling, employers are going to be thinking very, very hard about doing anything to GC people that might end up in front of a tribunal, because they will be getting their derrières handed to them on a silver platter.