But they also believe that racist and misogynistic speech should be heard
Yes, that is what free speech is. It isn’t “free speech for me but not for thee.”
women's concerns about male violence are overblown and might restrict men's rights.
ALL men, of whatever politics, believe women’s concerns regarding male violence are overblown. Because they do not navigate the world as women.
From what I’ve read, and not speaking of the alt-right here, but of the suburban Republicans who are the voters the Heritage Foundation came into existence to create, they do not want men in boys or men in women’s toilets or changing rooms, and they definitely do not want boys and men in girls and women’s sports teams. The men are coming at it from an urge to protect that is patriarchal in nature and, wrt to sport, with an understanding about fairness.
Be cautious about how very broad the paintbrush is you’re using.
At base, most people are going to resist, for very valid reasons, an ideology that insists they deny material reality.
It’s possible that the women seeking legislative assistance from Republicans in this issue may be saving the Democrats from themselves.
Remember that the US does not have a parliamentary system. It generally requires votes from different members of both parties to pass or stop legislation.
Bernie Sanders, right now, is the most admired politician in America. Why? Because he's been touring Republican states, listening to people there, and giving speeches drawing crowds if thousands. The right-left split is not as clear-cut in America and national media portrays. My very right-wing cousins, for instance, support Medicare for All, as do over 60% of Republican voters.