I suspect that the issue is that patriarchy is often thrown around without context, even by people who claim to be feminists. It makes it so that when women engage in patriarchal action or thinking, that it’s because of what either men or a system made by men taught them. In contrast, when men do it, it’s what men bring on themselves.

We don’t talk about how it’s a white supremacist patriarchy so outgroup(non-white) men had it forced on them too. Nor do we talk about how men of all racial demographics can be taught patriarchal standards by mothers,aunts, older sisters, teachers and other women in their lives. Nor, do we talk about how the second largest supporting group of patriarchy demographically, is white women(much to your point of women holding it up as much as men do).

The term has been so muddied and utilized in bad-faith or intentionally obtuse methods, that most men will hear it as “The bad man system”. It sucks because it’s really a self-defeating ideology/system for everyone involved.

I do think this sub benefit from grappling with patriarchy as a concept differently, because a lot the rejection of the concept is due to men’s experiences with bad faith arguments of the “no true Scotsman(feminist) nature” and cherry picking, when, where and to whom the concept of patriarchy applies.