What I am arguing is that using some magic invisible entities like "institutional bias" makes possible to prove anything you want - and by that, proves nothing at all. Instead of trying to seek the real causes of event, you just say "oh, it's those white males and their institutional bias, nothing to think about here". That's a lazy and nonsensical take.
> In the case of Republicans, we have a group of mostly older white men
Here you are purposely confusing between "majority of them" and "all of them, or at least all worth considering". It's like saying "Americans are mostly white". Demographically, it's true, 73% of Americans are white, but continuing the conversation while ignoring the other 26% would be pretty much not talking about America. Also, 50.5% of Americans are female - the majority. Would it make sense to talk about America as "the country of white women" and ignore all others?
> Republicans are over 1.5 times more likely than Democrats to say that women should stay home and not work
"Should" does a lot of work here. Does it mean "banned by the law from working"? Does it mean "if man marries a woman, he should earn enough to let woman not work if she doesn't want to"? Does it mean "one-earner families are better environment for children than two-earner families"? Does it mean "I would never marry a woman who wants to work"? Does it mean "this is the ideal role-model situation, which we will strive for, though in real life we often fall short of it"? Can mean any of those and hundred other things. What it doesn't mean though is "Republicans hate women, including women, who have internalized self-hate".
> But what it adds up to is that Republicans broadly support traditional patriarchal positions about the role of women, while opposing social changes and policies that most women would like, and feel gives them more freedom to live their lives as they will.
Are you sure most women think working some mind-numbingly mundane job for 8 hours every day is more freedom for them than staying at home with their children (or even without them, just doing whatever they like)? Surely, if the women is a CEO she would like to keep the job likely. Most working women aren't CEOs though. Are you sure some of them wouldn't feel more free if they didn't have to work?
Anyway, this argument is like 100 years late - nobody prohibits any women from working, and no Republicans that I heard of proposed any legislation that would ban women from working. However, if you think that not taking money from other people to pay for some woman's childcare needs is equal to "patriarchal position" - that's nonsense of the first degree. Nobody owes anybody else - men or women - to pay for their childcare needs. It has nothing to do with "patriarchal" - it's just that I do not have to give money to care for your child, because it's my money and not my child. Take your own money - or, if you have a decent job, your employer's money - and care for your child, nobody bans you from that. Trying to guilt people into giving you money to care for your children and calling them "patriarchal" if they refuse is just an abuse of terms. Presenting them as if they don't give up their own money with no benefit for them to benefit some stranger women as them hating women is plain disingenuous.