I agree women are usually straddled with a double burden in many homes. That's why I'm a stay at home dad so my wife doesn't have that problem. How would women running the household solve this problem? By making men take care of the kids? Would a heavy handed matriarchy be any better than a patriarchy in terms of fairness and top down expectations? Legitimately curious.

I think people just need to make firm expectations long before they have kids so everyone knows what to expect. And divide labor accordingly ahead of time.