Your first sentence suggests you're trying to argue, but I don't see where the contradiction is. Even the transmeds would agree with what you've written.¹
Trans people don't be trans in order to fight sexist gender roles. They're trans, and (often) fight sexist gender roles.
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Some pedantic notes that may or may not be relevant:
• The two different domains you list are influenced by different dimensions of biological sex, which can vary independently.
• A social gender, while culturally-influenced, seems to represent something fundamental about human experience.² Many societies' social gender categories do not align 1-to-1 with biological sex classifications, but 'most all of them have them. Meanwhile, race isn't seen outside racists and those influenced by racism: it is a recent³ invention.
• Some people's brains, for whatever reason, do care about the sexual characteristics of the rest of the body. This isn't much of a problem if that "preference" matches what the body is, but it can cause people significant distress when they don't.⁴ ⁵
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¹: Though transmeds would dispute the (perfectly valid) implication in your second sentence. "Born in the wrong body" isn't an entirely accurate explanation: it's more of a lie-to-adults, only required if you've got a stodgy old binarist model in your head that can just about conceptualise the existence of gay people. It's up there with "lesbian relationships are butch/femme": not something we need to be teaching children, though probably better than acting like LGBT people don't exist.
Brandon Ambrosino wrote a good article about the "born this way" argument: it goes into more detail, with anecdotal examples. https://www.bbc.co.uk/future/article/20160627-i-am-gay-but-i...
²: Disclaimer: I'm not speaking from experience. My intuition is that gendering is bad. From observing other people actually liking, and gaining value from, gendering themselves, I suspect there might be value in keeping gender around, for their sakes; however I have not really taken the time to square this with the clear and obvious harm caused by our society's obsessive gender-all-the-things-and-people tendencies.
³: Dating back at most 1400 years, with its European-coloniser incarnation probably not much more than 400 years old, and the modern eugenicist version less than 200 years old.
⁴: This doesn't necessarily even line up with gender! I know a cis guy who went on feminising HRT, and he says it made him a lot happier. (I don't have access to the inside of his head, but I'd be inclined to believe him.)
⁵: Personally, I don't see why someone should need to have a medical condition to get to choose what their body is like. I get the rationale, seeing as these are semi-permanent choices that one's future self might disagree with – but we could apply the same logic to tattoos, nose rings, or earlobe stretching. (Perhaps we should!)
Regardless, the fact remains that many people do have such a medical condition.