I think one of the reasons feminism gets such a bad name is because people seem to believe that it's anti men and that when we talk about the 'patriarchy' we mean that men are consciously involved in trying to keep women down. Men get defensive when faced with that attitude and women who know their partners are not like that get defensive on their behalf and dismiss feminism as having no relevance to them.
TBH before I had children I used to think like that too, despite believing in equality and knowing I was just as good as any man. The thing is feminists in the main do NOT believe that all, or even most, men are consciously and deliberately engaged in subjugating women (well a few notable misogynists are, but...). Most sexism/discrimination is done subconsciously, by women as well as men, simply because all it is doing is reinforcing the status quo. It's normal.
The patriarchy may not be a living, breathing thing but it does have a life of its own and it self-perpetuates. Patriarchy can be seen as simply another word for society. The name patriarchy is given instead of society simply to demonstrate that this particular society is male dominated. It is made up of individuals but it creates social norms that keep those individuals in check. And those social norms, because they have evolved over centuries (millennia) where the cultural dominance belonged to men, tend to support male dominance over women. It will take a long time to redress the balance even with equal rights legislation. As is the case with racism, which has been around far less than sexism.
The patriarchy damages many men as well as women. When it places derision on them for wanting to be a SAHD and 'failing to support their families'. When it places suspicion on them for wanting to work with children. When it doesn't allow them equal time to spend with their newborn child. When it portrays ridiculous machismo as ideal 'male' behaviour. When it calls sensitive boys 'sissies'. When it says "Oh men!" in that patronising way that implies men cannot manage to cook a meal/care for children/clean the house/cope when ill and are about as capable of multi-tasking as the average 3-year-old.
Feminism wants an end to these things as well as extra rights for women. In some cases they are two sides of the same coin and one will come as the direct result of the other.
Feminism is NOT anti-men. Anti-feminists OTOH want to keep men locked in their traditional gender roles just as much as they want women kept in theirs.