I often thought when gi seemed to be just taking off that it could never happen in a place or time without reliable birth control, and I think your comment shows why this is.
You can have alternate sexual categories for gay men in a situation like that, without too much trouble. Except that obviously it was not something that would produce offspring, which was quite important in many cultures. So even where you had these third gender categories, they didn't do things like marry in the conventional way. And in some cases these arrangements could be used to provide sexual outlets for men that did not involve the risk of producing children the society needed to deal with. So the sexual rules for a male third gender could afford to be relatively generous in terms of behaviour, so long as it didn't interfere with things like inheritance and producing a new generation.
But the fact is that in order to let women live like men, in the past, the only real option was celibacy for those women. And so any female third gender represented an end of the line for that woman's genetic material, and the whole concept depended on her having much more strict sexual rules than both men and women living conventionally.
But you are right, none of them is anything like what gender ideology in the west claims.