Well obviously it is false as a matter of fact. Anyone who does the slightest bit of research about transition finds a zillion cis(-ish) people who question their gender for any person who commits to transitioning, gender fluidity, effects of socialization, and a mountain of doubts and steps backwards in every trans person but the most poster-childish. Anyone who digs a bit deeper will find heavy philosophizing and introspection about how there is no "deep down" for gender or any other identity, the social construction of gender, the weird hangups and questioning about each step of social or medical transition, the hard choices between ideal gender expression and social pressure that makes the notion of real identity meaningless, and a bunch of people who detransition and sometimes kill themselves.

But someone who does not want to the research, and would even prefer to stop thinking about the creepy stuff as quickly as possible, is going to need a simplistic caricature, preferably one that doesn't take apart the concept of little neat gender boxes at all. A mainstream one is "A man decides he'd rather be a woman, and becomes one". (Another is "A man decides he'd rather be a woman, but of course he can never be".) gwern basically seems to use that one. It's not a very good one - it casts trans people as inexplicably making a weird choice, it misrepresents pre-transition people even worse than mine, and in basically all instances it's too focused on physical sex. "A woman is misclassified as a man, finds out and corrects it" is a more useful approximation. It encourages approximately the right behaviors (e.g. shutting the fuck up about birth names), and is closer to the motivation of transpeople than the "choice to change" one.