Right, but you are using the -archies in a way that from my understanding is not correct.

A patriarchy, as applied to the broader social structure of a civilization, has a higher concentration of men as one looks upward in the hierarchy.

Nothing more.

Definition from Why Men Rule by S. Goldberg:

This is why the appearance of Theresa May did not suddenly make England a patriarchy. She didn’t alter much the concentration men in authority.

It’s not a system that favors men or disfavors women necessarily, it is that the structure of power as you look up is more male.

That structure can certainly enact rules that favor women, or recognize women’s roles… even pivotal ones in the lower strata. It can choose matrilineal family structures… but if it doesn’t alter the upward concentration of men… then it’s still men deciding families are to be structured matrilineally.

I believe that is how this works, anyway.