Gender as a concept distinct from sex is a way of describing an existing social phenomenon. Gender, though traditionally tied to biological sex, changes over time. What we define as "masculine" and "feminine" traits change as society changes, and can even differ between societies, despite the fact that male and female chromosomes have not changed. Just because no one had described this distinction before the last century doesn't mean it isn't legitimate.
The terminology of the debate gets confused a bit because while indeed the expectations of people’s behavior, dress, etc. based on their sex is indeed a social construct and has been probably since the beginning of humanity, that wasn’t what “gender” meant until very recently. As far as I can tell, “gender” and “sex” were completely synonymous until 50 years ago, even in niche academic communities, and until maybe 10 years ago among the general public.
I will agree that the terminology could probably be better. But my original point is that this is a pretty high-profile issue and has been for a while, so anyone taking a hardline stance of "I'm gonna deny trans people until someone makes a good argument why gender and sex are different" is willfully ignorant at best.
All categories are arbitrary distinctions that someone made up at some point, including the categories of biological sex, race, and your birth name. At the very least it would be bullying behavior for me to single you out and call you a different name or different racial group to what you actually are (as defined by the constructs that society has agreed upon).
Sex is an emergent property of multiple genetic networks. This doesn't mean that it doesn't exist or that it is not bi-modal, but it is far from simple.
The fact that we don't go around performing genetic testing on people before we interact with them also raises questions about whether sex itself matters, or whether there must be different definitions for sexed categories dependent on policy purpose. (e.g. based on genitals when the concern is sexual violence; based on fertility for the purposes of reproductive medicine; based on gender performance; based on hormone levels, etc.)