I’m not accepting that as an axiom. I’m asking you (and others) to explain what gender is, in this context, non-circularly. So far, I haven’t gotten a definition.
Basically, it's a set of socially prescribed behaviours and expectations based on sex. Men wear trousers and are strong, women wear dresses and are graceful. That sort of thing. There's a few different concepts mixed up in the idea of gender, but it's effectively the social aspects associated with sex.
If so, is your original “(1)” point entirely objecting to people saying “I feel I was born in the wrong body”? I, and I think others, were certainly reading it as “trans people aren’t a real thing”, with the implied policy implications that carries with it…
The argument isn’t that sex and gender are entirely unrelated concepts. Just that they’re not inherently equivalent.
Edit: I think fundamentally, if you don't object to treating trans people decently like using their preferred pronouns and name, or getting surgery, I don't think it matters too much. I'm not an expert on the definitions by any means, my primary concern is opposing justifications to mistreat trans people, so if it's just a matter of terminology I guess my only suggestion is to read into what the relevant fields of study have to say if it's something you want clarity on.
You’d be free to argue that this isn’t an inherent property, I suppose, given initial probable-fluidity. But since it seems to settle into being a largely fixed part of your psyche before the point you’re likely to have permanent memories, I’m inclined to view that as a meaningless difference.
It’s at about the same level as arguing whether sexuality is an inherent property, I think? It’s another of those “it might be hardcoded or it might be early-development cultural, but it’s basically impossible to change it so…” things.