So... your argument that we do live in a patriarchal society is (1) we get surnames after dads (I mean, really, who cares), and (2) the unsubstantiated assertion that men are the "head" of the household (what does that even mean? that they spend less time with kids? that they earn more money so the wife can spen more? that they're less likely to keep the kids in event of divorce?)? Oh, well, now you've convinced me, of course it's a patriarchy!