As I said throughout this essay, patriarchy is fundamentally a dominance hierarchy. If women behave in patriarchal ways (and they often do), you get the same outcomes. The two opposites here are not really men and women, they are patriachal dominance hierarchies and partnership-oriented social structures.
"The first sentence of the New Scientist quote is important because it speaks to a central aspect of patriarchy — social stratification. This was the first time that class distinctions or significant hierarchy came into widespread use, and they were maintained by force or the threat of violence. This is an important aspect of patriarchy — a social system that embraces and rewards ruthlessness and a might makes right mentality. This is distinctly different from a time when the wellbeing of the entire clan or settlement was a primary survival strategy."