If anyone is “in charge” you’ve still got a dominance hierarchy. Matriarchy is not in any way, shape, or fashion preferable, but it’s not the only other option. For instance, there is a movement in part of Syria towards a more cooperative and truly democratic society. It’s not perfect, but it does indicate that a dominance-based hierarchy is not inevitable.
In the heart of Syria's darkness, a democratic, egalitarian and feminist society emerges
The most amazing thing about Rojava is that hardly anyone knows it exists. We hear plenty about Syria the battlefields…
Patriarchy is still the appropriate word for the dominance hierarchy that we have in the US (and in most places) because in our culture men hold the bulk of both political and economic power as well as moral authority in society.
Traits that are considered masculine like control, competitiveness, and stoicism are more desirable for everyone than traits that are considered feminine like empathy, nurturance, and cooperation.
“The core cultural ideas about what is good, desirable, preferable or normal are associated with how we think about men and masculinity.”
However, we’re all victims of a patriarchal system. It harms men, women, and others, albeit in slightly different ways, because patriarchy is a system based on stratification, coercion and control. That means that you always have to be jousting for your place in the hierarchy, and that for men in particular, if you show any vulnerability or perceived weakness, you are likely to be eaten in the dog-eat-dog dynamic.
I absolutely do not let women off the hook for upholding patriarchy.
These are some of the other ways that women uphold the patriarchy:
Maintaining gender binaries; enforcing what is considered appropriate for boys or girls, rather than letting each child be who they naturally are.
Obsession with female appearance and body image; both their own and that of other women.
Shaming men and boys who show emotion or vulnerability.
Participating in policing female sexuality by using terms like slut and looking down on women who are sexually confident and self-expressed. This includes, but is not limited to, the denigration of strippers and sex workers.
Upholding the traditional nuclear family as the ideal, rather than embracing a variety of nurturing and supportive family styles.
Not acknowledging that patriarchal norms limit and harm men too.
If we are ever going to have a real chance of creating a more kind, inclusive, and gentle society, women need to understand how patriarchy not only works against them but how it works against everyone but a few elites at the apex of the power pyramid.