Actually women's rights were generally thought of as 'left wing', before the difference between right wing and left wing misogynists became a blur.
I completely agree that religious crackpots like Douglas Wilson want us all to be living in the Handmaid's Tale, and they have been given a head start in the US by all the 'progressive' organisations who can't talk about abortion and contraception as women's rights issues for fear of hurting men's egos.
You cannot protect women's rights like access to birth control if you cannot explain what a woman is. Covid had a sex specific impact on maternity and post natal care, access to the morning after pill, access to fertility treatment, but according to some 'progressive' groups these are just random things that could happen to anyone so any kind of analysis of the impact on women is exclusionary.
The Equality Act in the UK is under attack both from people who want to repeal it completely and people who want to remove sex based discrimination from the act.
Honestly, I cannot tell the difference between somebody who thinks women should meet gendered expectations and behave like a tradwife and somebody who thinks a woman who does not meet gendered expectations isn't a woman.