There's no such thing as an atomized patriarchy. A man who attempts to assume the privileges and duties of a patriarch in his home while living under a societal gynarchy won't have any leverage to enforce his edicts. The right ZFG attitude can get him some ways there, but one slip and he's doomed.
A patriarchal society is necessary for the flourishing of a patriarchal family unit. White men had this, until roughly 1970. Knowing that the society would defer to White men gave them a lot of power over family and cultural affairs, and this resulted in a cornucopia for everyone.
If White male leadership stops at the front door, it won't last long behind the door.
