> many cultures across the Americas and Oceania had and have non-binary systems,

As I understand it, this is because these cultures had deeply sexist ideas about how women and men should behave, so they created additional categories to shovel everyone who didn't conform into. In practice this tended to mean that gay men would be placed in some sort of "non-man" male category. So while sexuality and gender are different things, in practice they end up linked through this mechanism of othering.

Right, so then the missionaries with deeply sexist ideas about how women and men should behave came and shoved nonconformers in a binary system.