Father Time said:
I thought it was a society where power is specifically handed down to men and only men, like those old monarchies where the throne would always go to one of the King's sons (if he had one).
I think that is
Patrilineality that you are thinking of, that is about inheritance.
An -archy is more generally about the social hierarchy.
(And a -cracy is about the way the government and political system gets organized.)
Father Time said:
That's the literal definition but not the one I've seen most feminists use (they even tell me they don't mean the literal definition). Like when they say patriarchy hurts men. You tell them that men get higher sentences for the same crimes as women and they say it's the patriarchy's fault.
Those that you hear are not the definition, but the added explanations about HOW they believe the patriarchy works.
If someone tells you that they believe the USA is heading towards a dictatorship, and then start a rant about conspiracies, the proper reply is not that you "don't believe that dictatorship is a thing". It is a given, that the specific conspiracies they have listed are not a fundamental part of the word "dictatorship", just of their specific usage of it.
And to answer your specific problem with patriarchy, I think what you are missing is that just because a society is "ruled by men", doesn't mean
A) that it is ruled by every single men, (as opposed to a handful of men ruling over most men and women)
B) that every single person benefits from being considered "in authority". Edward II of England was the Monarch in a Monarchy, yet I bet he would have rather chosen to be some obscure peasant, by the time he got a red hot poker inserted into his anus. Hitler might have been Führer in his own social system, but by the time the Soviets were coming for him, he would rather have been a jew in Treblinka, just getting liberated.
If you think that right now we are not living in a patriarchy, try to imagine the most patriarchal system you know. Medieval Europe, islamic Sharia law, whatever. You might notice, that even there, men can be harmed by the very same logic that places them at the top of society. Men are supposed to fight in wars, men are supposed to be chivalrious, men are supposed to be more legally accountable while women are like children under care...
So you could either conclude that there is no such thing as real patriarchy, (not even in the countries where women weren't allowed to own property, or legally defend themselves, or choose their husbands), or more rationally, that all patriarchy can benefit women and harm men on some smaller scales, even if by and large this is done by granting men more power than to women.