One of the reasons I've never self-identified as either culturally "progressive" or "conservative" is because I've always held a mix of beliefs that make both labels inaccurate. I've hated gender roles ever since I was teenager but also reject the idea that gender is a choice and some of the silly "gender identities" you find in progressive circles, like non-binary he/him people.
The problem with gender roles is that sex is almost never actually relevant to anything other than, you know, making kids. Justifications for filtering or grouping people by gender are usually associated trait fallacies.
Often, defenders of gender roles will bring up some sort of statistics to argue that males and females naturally have different minds and that this should be embraced. These arguments are always fallacious for a number of reasons, such as the typical problems with statistical arguments, and the fact that it's a self-fulfilling prophecy. We were all taught gender roles as kids, even if only implicitly, and people act the way they're raised to think is normal for them. Thus, any statistics you're using to show that males and females naturally have different psychology is adequately explained by the prevalence of the belief you're using them to argue for. (While many political "liberals" claim to oppose gender roles, they usually still welcome the idea that there are biologically-caused differences in psychology between men and women; so yes, it is a prevalent belief.)
It's actually for the same reason that I disapprove of "gender is a choice" folks. I think that wanting to "identify" as a gender other than what you physically are shows they are still not properly separating gender from personality. I think they think separating "gender" from "sex", where gender is a choice with no wrong answers, solves the problem, but it doesn't, because then they define their identity in terms of the same language used for sex. This is misguided. Personality doesn't need to be tied to gender. You don't need to be transgender to be gender non-comforming.
This should not be taken as a condemnation of transgender people. Mind you I used to think gender dysphoria wasn't legitimate and that people were only uncomfortable with their bodies because they weren't properly separating sex from personality. I've been proven wrong though, because I've seen many of these people make exactly the same points that I just did, get furious at people who say being trans is a choice, and say that they would choose to be cis if they could. They really do have a legitimate medical condition, not a sexist bias.