So I don't entirely disagree with you here - what I'm trying to say is that it's more complicated than that. I just replied to neurotransmissions mentioning this, but there are current and historic cultures that have more than two genders. Hardcore historical traditional Jewish culture has... 6 I think (?), not that they really line up with how we think about gender. India broadly has Three cultures including Hijras, though their official government recognition is very new.
Basically what I'm saying is that humans have been trying to describe and decide where to put trans people for a very long time. This is why I have a problem with the question "would trans people exist without gender as a construct", because it misunderstands the purpose of that same construct. Each of these current and historical gender constructs, including our own current one, are just a tool that we are using to try and figure out how we fit into the world. Trans people are often inconvenient because of how we complicate that tool's functioning.
The construct itself is meaningless because we created it to try and describe what we see - It is dependent on us, we are not dependent on it. So, Trans people do exist without the construct of gender, that's why different cultures throughout history have had so many different gender constructs: They're trying to describe everyone, including us.