I believe the gender movement is heavily linked to capitalism, and part of the reason it's being pushed so hard in the US in particular is because of our ultra capitalist system. It's making thousands of people dependent on big pharma companies for the rest of their lives, and TiMs also boost capitalism by buying unnecessary products to aid their transition.
I also believe capitalism has exploited women in other ways throughout its history, including forcing them into unpaid home labour, and harming lesbians by propping up the ideal of the nuclear heterosexual family. Capitalists also have an incentive in causing women to be insecure about their bodies, so they can make us spend unnecessarily on things like makeup, razors, and cosmetic surgery. If I have my history right, the entire thing about a lack of body hair being valued in women was created by businesses to sell razors to women.
Thus, I believe, if implemented correctly, socialism could be a benefit to women, and would have less of a place for gender ideology.