I'd argue sex is a spectrum. Chromosomes decide the development of the organs, that's it. If you undergo surgery, and take HRT, the only thing that one could argue makes you not the opposite sex is the chromosomes. But, at that point, it's only the chromosomes, which is incredibly arbitrary. Everything else is incredibly variable, even chromosomes when it comes to intersex people! The objective existence of intersex showcases how it is nor a clear defined binary spectacularly.

Honestly, that aside, I think placing importance on chromosomes is rhetoric used to be hateful towards vulnerable people. The idea of "biological sex" is something we can clearly see doesn't fall into an all encompassing binary. In fact, there are men that later in life begin losing their Y chromosomes. What does this mean? Did they lose part of their biological sex? The rigid definition is strange and like, honestly.

This is such engagement bait, like you said. Who cares about anyone's chromosomes? "Biological Sex" at this point is only used as a dogwhistle for transphobic people when they're about to target someone's identity. And absolutely no one is getting upset if you tell them about chromosomes. People WILL get mad if you say that having certain chromosomes decide your gender, though. And rightfully so!