I feel the difference between the biological imperatives of the sexes is the real perpetrator, rather than a "patriarchy".
Men and women have different preferential 'moves on the board' that benefit or cost them in different ways because men and women are different when it comes to what they must do to survive in the world, due primarily to their biology. Genders aren't opposites at all; they do just have different roles because nature has forced that upon us, and then through sexual dimorphism and the extension of adaptation through culture, ensured that those acting like the sex they are 'not' are going to come up against obstacles. The human race has a strategy - a complex but effective one - for survival that meant identifying as a gender and then adopting the role of that gender made things run more smoothly and probably led to our becoming one of the most successful species on the planet. The fact we prioritize the safety of women and children for example, is one of the reasons we're so successful. If we truly were ruled by a patriarchy that put men in all things before women, we'd have died out long ago, because the human female - especially the pregnant one - kind of requires safety and resources and help from other human beings to successfully birth and raise a child. This 'patriarchal gender structure' needs to be ditched because I'm pretty sure we don't live in a world run by women-hating men in which women are not cared for and not protected and not provided for. Women get an awful lot of provision and help and concern for their safety in all societies. The traditional definition of the Patriarchy is that it exists only to subjugate women to benefit men. In a world like that, why would any man choose to help a woman? Or give her anything other than the absolute bare minimum for basic bodily survival? It's clear we do not live in such a world at all, because such a world would not last long.
So I think I could answer this question at some point, but when we clear up this Patriarchy business and start understanding that the way the genders treat each other is ultimately biologically-based, and comparing ourselves on a scale that probably isn't indicative of much at all isn't going to give us much useful information about ourselves? Honestly I don't think we're going to move on in this world until we understand the real reasons behind why sexism and sexual stereotypes evolved at all, where they succeeded and where they failed, and why. It's so persistent for a reason.
