Perhaps it's the difference in how leadership often plays out differently in patriarchal cultures. As you say, men tend to organize things to suit them, which isn't actual leadership, it's privilege. For women, leadership is more often about making sure things get done than aggrandizing themselves. Not always, of course, but often. My husband and I have a pretty great relationship and very much a partnership, but I do notice that left to our own devices, he tends to take on the tasks and chores that appeal to him in the moment or that he feels like doing and I tend to do what needs to be done. He'll pitch in to do other things if asked but it doesn't seem to naturally occur to him. He's used to suiting himself. I'm used to thinking about the larger picture.
Matrilineal cultures believe in balance between the genders, patrilineal cultures don't tend to care much about anyone but men. Women and children are both expendable and interchangeable. Of course, this too is on a continuum depending on the culture, but is at least partially true in most patriarchies. Matrilineal cultures tend to look at the culture through the eyes of a nurturing, supportive mother, the people who tend to be most revered in those societies. We already know about patriarchies - where the most important thing is to climb the ladder of the male-dominated pecking order and everyone else is an antecedent.
I definitely need to write a full story on this. Thanks for the writing prompt. 😎