I'm in no way knowledgeable regarding this but does gender politics not further gender distinctios? Particular trasgender politics seem go reinforce gender roles rather than diluting them. It would seem wiser to modify your body and personality however you wish and adhere to sex, without worry of defining a gender. I suppose situations where the sexes are separated is where gender politics start.
"Second wave feminism" (roughly, the 1960s through 80s) tried to improve equality by minimizing differences. It was understood that gender was a social construction, and so could be re-constructed. It was always seen that women could choose traditional social roles if they wished. But it was expected that most wouldn't.
But there were problems with that approach that didn't appear until later. Transgender people were being ignored. It's not that they didn't exist; as has been noted, some of the most prominent activists at Stonewall (a key event in gay American history) were trans women.
Trans women are women, and denying the importance of gender constructs undercuts the work they put into their true gender. In second-wave feminism, transgenderism could be seen as gender essentialism, that gender wasn't constructed. That risked marginalizing women who had fought not to be dismissed because of their anatomy.
The approach of modern feminism is more intersectional. It incorporates all kinds of different marginalized people. This is in contrast to the explicit white racism in first-wave feminism (the ones leading up to the right to vote) and the subtler racism in second-wave feminism (which focused more heavily on issues facing white women). That approach also incorporates gays and lesbians, disabled people, and lots of other ways in which people are harmed by the culture that people persist in solely because it's "traditional".
There are a few "trans-exclusive radical feminists" clinging to older approaches that don't admit trans women, but there are so few of those the the acronym "TERF" ends up being applied to almost any anti-trans woman.
Now gender is seen as a spectrum with many axes. It doesn't matter that some trans women want explicitly to identify with femininity, even as cis women are liberated from its strictures. A lot of people don't fit into either binary or anywhere along that axis, so there's less worry about reinforcing gender roles.
Sorry for the length of this post; I'm summarizing a ton of messy stuff (and, I'm afraid, not doing it entirely well). I was asking the same questions you are, not that long ago, and it took me a while to get past the assumptions that I learned as a second-wave feminist. This is my best understanding of it.