I agree: but I don’t know if you can separate those two things, ultimately.
If people hate (or are even merely indifferent) to women, then it’s unlikely they’ll take the politically difficult steps necessary to wipe out misogyny from the law as well.
Side-note: In the year 2020, I actually think we’ve done a pretty good job in America of achieving formal legal equality between men and women in most respects, with the notable exception of reproductive freedom.
There are all sorts of barriers to obtaining an abortion in many states, and a significant minority of conservatives still want to drag us even further back to the stone ages, all the way to no abortion at all.
That’s why I’m not so sanguine about misogyny, racism, homophobia, transphobia, etc. with the disclaimer that it’s “just a personal belief.”
Change typically begins with changing attitudes.