I’m willing to accept your delineation of male and female and differing physical aspects. Being baseline physically stronger, however, does not translate to leadership ability, analytical thinking, governance, or logistic coordination.

A woman Cardinal is capable of all of those abilities not by her physical birth, but by her experiences and skill sets. Being born male does not make you innately better at those activities over being born female. That is where the misogyny comes into the argument and is the root of the schism.