You refer to Genesis 3:16 as ruling out patriarchy whereas my reading of chapter 3 suggests the opposite. Surely the fact that Adam (man) and Eve (woman) were cursed separately and differently by God meant that God was giving them a blueprint of the social system mankind was to live under. Man isto work by the sweat of his brow whilst woman is placed under his domination and because having lost their total self-control, she is condemned to multiple, painful birthings.
You have made a basic assumption that patriarchy is wrong without showing why.