In JP's justification for orthodoxy's opposition to gender equality, he initially cites 2 main reasons: that women are prohibited from attending Talmud lectures, and that wealth is held by men.
First there is a logical fallacy here (asserting the consequent). Women don't hold wealth and can't learn Talmud? That's the point, because men have been (unfairly) dominant historically, women have been kept at a disadvantage, but that is not a reason to maintain it. Jews adopted male dominance from surrounding cultures (as they did many other things, including the name of their god El). It like someone saying in the 1800s that blacks shouldn't be allowed to vote because they had been slaves, and they thus are too uneducated.
Furthermore, that women aren't permitted to learn Talmud is a particularly severe and neo-conservative invention of the Litvaks. It is based on a number of Talmudic saying about women being simple minded and that teaching them Torah is a waste of time. But I suspect that the Litvak ultraorthodox don't really believe this anymore, but need to keep their women ignorant and thus keep them in their place. It is certainly not the view of mainstream orthodoxy and certainly not "Judaism" that says this.
JP also disingenuously uses biological arguments about the strength of males. He is a hypocrite as he has written many times about problematic human nature and the need for morals to tame or overcome them. Suddenly, because males can outfight females, that's a reason that men should make decisions? Sounds kind of pseudo-Darwinistic to me. Yes, on average, men are better at certain things and women at others. But that doesn't mean that they should be forced into a role.