Okay, I confess: we have so little honest, trusted, hands-on information about old institutions, I just snap to assuming the worst about them even after adjusting for less decay.

You can't have patriarchy without the father having greater formal political and legal power than the rest of the family.

OK, what if "the rest of the family" is somehow weak/timid/socially clueless/foreign/under-networked/from a disliked minority/whatever, and can't bring informal/"soft" power to bear in a dispute with the father? Seen lots and lots of times in literature! Works with the wicked stepmother and the spineless father, too. I fear some kind of Stepford Wives shit, but replicated with Singaporean efficiency!

which of these has the most obvious incentives for good outcomes for children themselves

Obvious counterpoint. Unless it's a TDT-using family (and we don't see much practical TDT used in real life... besides the evolved pseudo-TDT of religious/Universalist ethics, that is), every family has incentives to have its children compete and beat other children in zero-sum games. A big church or a state have incentives to discourage zero-sum games for all children, and promote cooperation instead.

And that does happen in practice, I think: most everyone who lived in the USSR would agree that its brainwashing of children was benign in that particular area - teaching cooperation and suppressing zero-sum games. That was only the official intent, of course; policies to that intent might have been as inefficient as everything Soviet.