I didn't get that from his essay. He wasn't saying the state has moral authority, but parents shouldn't assume it because children are free moral agents in the end. If that were not true, children would not be fleeing these religions in droves like they are as soon as they can leave their parents' home. There would be no ex-Catholics, #exVangelical would not be trending, the Amish would have 100% retention, and LDS young women would not be desperately escaping polygamist marriages in such high numbers.

It just doesn't work. You cannot control the thoughts and moral beliefs of another human being. Trying to do so will only damage the relationship between you. That goes for institutions too.

So I don't get where he implied anywhere that public institutions should have moral authority over children and parents shouldn't. He said parental thinking that they own their children is flawed.

There's a world of difference between children learning about a topic, learning what it is, the history, historical and current thought around it, how did effect people and determine their future, etc and trying to force them to accept moral judgements about such things. Yes, even all the thorny topics. One could argue that it is impossible to make sound moral judgements about said topics without knowing those things. Sure, you can make moral judgements, but they won't be sound, reasoned ones.

I see your point too, that a public institution can 'go bad' too. And they do, usually because the decision making body becomes exclusive to the broader public and therefore biased by their own world view they then try to out upon everyone else. Though, again, as the author said, you have the choice to remove your children from public education and teach them as you see fit. Right or wrong, that's the compromise and the law. You do NOT have the right to foist your own moral code upon the public at large.