You are mixing things up from what I said. Societies can be egalitarian, patriarchal, or matriarchal. A long lived egalitarian society isn't disproof of greater social structure stability in patriarchal societies than matriarchal ones.
Back to the repeated prisoners dilemma. Cooperate first is the ideal outcome. I'm pro cooperation. The society that is really cooperating is egalitarian.
You may desperately want it to be another way, but the high time preference win for men is to use their power to take control. I take control and now I get more power and control today. The long term costs are fuzzy possibilities like what if a woman was going to invent something cool but now can't because me and the other men took the resources and control away from her?
Egalitarian societies should be workable long term given the right conditions. Matriarchal society is just a different but still horrible place. Because feminists are the winners they still cheer for matriarchy because they get to be the winner.
You may disagree, but try being a man doing a service call in an all female business sometime. Crassness, leering, sexual harrasment, cruelty when they get turned down, I've been through all of it. Women definitely need men around to remain civilized every bit as much as men need women.