Patriarchy is a social system, although I agree that it’s still a bit of a loaded word, which is why I most often pair it with dominance hierarchy (which is a whole social pyramid that is about way more than men and women). I agree that toxic masculinity is too fraught at this stage to be useful.
“This is where most of the traits that have been identified as toxic come into play: aggression, bullying, violence, anger as the only acceptable difficult emotion, coercion, and extreme self-reliance that fails to value cooperation and inter-dependence, to name just a few. These are traits that anyone might exhibit and they are not native to males. These are traits that patriarchal society has demanded of boys and men as a part of participating in the dominance hierarchy all around them. Men who have embraced these traits as a way to demonstrate masculinity are not themselves toxic; they are simply purveyors of a toxic philosophy. They’ve been doing what they’ve been told they should by a warped and destructive system.”