But sex and gender are different. Sex is biological, so your point regarding chromosomes above applies there. You can turn a penis into a vagina but would still have male genes.

Gender is characteristics that a society or culture delineates as masculine or feminine. So while you can't change your biological sex, you can change gender. Which is what trans people do.

I also don't think Bruce Jenner is the ideal example of trans people. He came across quite strange in his interview, or at least in the clips I bothered to watch.

Like, where he wants to be a woman, but is still attracted to women but wouldn't then consider himself a lesbian. That's the opposite of what most trans people would think/how they identify.