If anything, the word "patriarchy" isn't used enough and I disagree that it's bandied around without thought. (Obvs some will use it without knowing too much about it.)

Your sentence - "The point of feminism is that women who don’t want that, don’t have to, won’t be pressured by society to do that and more." illustrates my point perfectly.

"Pressured by society" - and what kind of society is that right now? It's a patriarchal society. It's still run by mainly older, white men and yet for some reason, not enough people realise it's a patriarchy.

Perhaps if we start throwing the word around more frequently, more people will understand what a truly unequal world we still live in, whether it's being Mansplained to, or denied an education.