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.
As someone from a former communist country, it's absolutely dumb and anyone living in the west wishing for communism has absolutely no idea what they are talking about. Also every single gender special TRA has been a commie or socialist, so it's definitely not linked to capitalism. Women have been marginalized in every single economic system to date. The only decent(ish) thing that came from the Soviet Union was free childcare but that comes at the cost of getting your kids pumped full of propaganda and ideology at a young age. Imagine the only childcare options being heavily TRA and unquestionably for the only party in the country. It's already starting to be this way in the US so definitely a big no thanks.
Contrary to popular western belief, communism and socialism does not result in people's needs met and people just getting free stuff left and right. It means everyone gets the same trash quality thing and working non stop. You can't get extra nice things if you work harder, only if you are part of a corrupt system.
My whole family and countless others worked so hard to get to the US and always praise it and the opportunity it provides. Sadly the US is far from what it was and the new generation is no longer better off than the older one, but it's far better than in communism.
I really encourage anyone who is into this ideology to just seek a wide variety of primary sources and experiences from people who actually lived in communism and different regimes. Learn the reality of what it is, and then decide if you want to pursue that.
EDIT: I also wanted to add that I also laugh at conservatives who think every public good is evil communism or socialism. I am very pro things such as social security, WIC, etc (but never UBI). You would not have anything like that kind of support for people who need it in a truly communist country.