It is not a red hearing at all unless it is used to displace another message. So if you were talking about class warfare and your message was drowned out or disrupted by a message of destroy the patriarchy then you could say it is a red hearing.

Sexism isn't just a thing, it is "the" thing to be honest. For example, the leader of the KKK commented over a decade ago that he would rather vote for a black man (Obama) than a white woman (Hillary). Let that sink in, one of the most racist people is still more sexist than racist.

You bring up the fact that people in control use sexism to manipulate people. For sure they do just like they use racism. The crazy part is they might not even be racist or sexist themselves but they know it resonates with people so they will use it for their propaganda.

I think you make a great point about a disconnect that those who are against the patriarchy may have with class issues. On the other hand you could also tie the patriarchy to issue like transgender rights through rigid enforcement of gender roles and fueling genocide with hierarchical structures.