Wait - you think that all intersex conditions
still fall into the gender binary?

You're going to have to walk me through that logic. If they are all either male or female, then
they're not intersex. Because that's what "intersex" means.
Intersex is a bit of a misnomer. The preferred term nowadays is "differences of sexual development" or DSD. I understand they would prefer "variations in sexual development" but the problem with that is that it resolves to "VSD" which is the usual abbrieviation of an anomaly of heart development (ventricular septal defect). So DSD it is.
I think the term "gender" should be avoided in this thread. We are not talking about feelings in people's heads here, we are talking about the physical reality of sex and sex organs.
People with DSDs are all either male or female like the rest of us. XXY is Klinefelter's syndrome, which is a condition of males. X0 is Turner's syndrome, which is a condition of females. And so on. In only about 0.2% of births is there any ambiguity at all as to which sex a child is, even though a DSD may be present. (Just as someone with a VSD actually has a heart.)
In the 0.2% of cases more specialised investigation including chromosome typing and hormone analyses are necessary to work it out, but it can be done. Always.
The so-called "hermaphrodite" situation, ovotesticular syndrome, is the nearest you can come to the fabled "intersex", but even there the person is not a third sex, and is not both sexes. Nobody has ever been born who can produce both sperm and ova. It's more a case of a male or a female infant having some anomalous tissue which has formed as early germinal material of the other sex.
There are only two sexes, everyone is either one or the other, and the vast majority of people with DSDs (particularly the ones where there is no doubt whatsoever which sex the person is) get quite ratty when activists co-opt their medical problems to declare that they are in some way not "real" men or women.
People with DSDs occasionally transition, though not at a higher rate than the rest of the population as far as I know. I met someone on Twitter the other day who said he was XXXY, which is a form of Klinefelter's, a condition of males, but he was living as a woman. He was extremely clear that there is no third sex and that his biological sex is male though.
On the other hand Erik Schinneger, who was raised as a girl in the 1950s in error, "transitioned" to his true biological sex of male once this was discovered when he was about 20.
So there are anomalies, but the anomalies are not a spectrum between the sexes, they are anomalies of one sex or the other, and there is no third sex and no third type of gamete.