But aren't patriarchy and misogyny fundamentally related? I mean, are there any societies with strong patriarchal norms but very little misogyny?
I do not know of any societies with strong patriarchal norms and very little misogyny, if they do indeed exist.
Patriarchy in cultures is without fail about the control of women. All niceties around protecting them boil down to treating them as property, as chattel. You fundamentally don't have an equal situation when one gender is rendered property of another.
ergo misogyny