I think you’ve got it wrong way around. Patriarchy is not responsible for these behaviors, these behaviors are (in part) responsible for patriarchy.
Like any mammalian pack animals, males instinctively seek to dominate other males, females seek to dominate other females, and males seek to dominate females. (As groups overall. Individuals vary.)
The fact that children do the same even before they are aware of social conventions, and that virtually every culture on the planet follows that pattern shows that it is far more nature than nurture.
Fortunately, humans are not slaves to instinct, and we are capable of modifying our behavior so your points are still well taken. But we shouldn’t fool ourselves about causes.