Expendability theory: A possible evolutionary explanation for Patriarchy

You have two societies - Society X and Society Y.
Society X is an Amazon matriarchy in which woman fight and risk danger while men stay at home and look after children.
Society Y is a patriarchy where men fight and risk danger and women stay at home.
In society X, because only women fight and risk danger, only women are killed, the population of women therefore plummets. Now, because mothers can only produce so many children at a time, the population cannot recover quickly.
In society Y, because only men fight and risk danger, only men are killed and the population of men plummets but because fathers can produce many children with different mothers, the population can recover quickly.
In other words, men are more expendable than women because you require less men than women to repair a diminished population and, because they're more expendable, they're the ones who would risk being killed in fighting and dangerous activities.
So a possible evolutionary explanation for why patriarchy existed in traditional societies is because the population of a patriarchy could recover more quickly from the damage caused by fighting and risk taking than the population of a society where women fought and took dangerous risks.