There are a few problems here:
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you are using "trans" in a way that might be a bit more narrow than I was meaning, i.e. you seem to use it to mean a person who was born one gender of the binary and who transitions to be the other, rather than as the umbrella term that encompasses gender non-conforming people, non-binary people, cross-dressers, drag performers, as well as people who transition socially and/or medically.
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You are splitting hairs on the gender binary as if we should be considering it as a separate from the anti-trans gender concept. Instead I think there is a dominant gender ideology that both thinks of gender as binary and is trans-denying.
Here were some of your statements I was responding to:
["two genders"] does not at face value refute directly gender transitions,
the false idea that the adage “there are only two genders” has anything to do with the existence of transgender people
I think to the contrary that the "there are only two genders" does relate to transgender people - particularly trans people who do not fall strictly within that binary or who fall somewhere on a spectrum between two poles, but even for people who do fit one one pole or another of the binary, transitioning from one pole to the other often gets treated as wrong / impossible / invalid under the gender binary concept because it's baked into the concept that the binary is rigid - you can't "change" your gender, and you can't be in between, which is how people see and understand gender and trans people under that gender concept (as trans people "changing" and as being in-between, esp. during transition, but even after).
So while I don't disagree with your point that third-gendering can be invalidating to some binary trans people (and the author was being lazy / over-simplifying / ignorant and thus could have done better), I think it's a little mistaken to focus so much on this small mistake and to direct that anger towards the pro-trans author (your ally) when the larger context is what matters and is still accurate - the dominant, oppressive gender concept is anti-trans, and the author is right to call out the anti-trans policy as anti-trans.