Private schools would be fine. Why do we have public schools in the first place? Wasn't it so that even the poor could get a rudimentary education? Why does religion or morals have to come into academics at all? Isn't that for your church to handle or do you need it everywhere?
The public schools were NOT created so that "the poor could have an education." "The poor" included Abraham Lincoln, who did OK, as did many others.
The public schools were begun specifically to make all those immigrants learn Americanism. Ask John Dewey.
Oh, and by the way--that included specifically excluding denominational differentiation; thus, Jewish children were specifically NOT instructed in Judaism, Catholics specifically NOT in Catholicism.
As long as the public schools stuck to the universal moral code, that was OK, within its limits. But then, the public schools went off the moral code.
It's not a coincidence, by the way, that only 50 years after prayer in schools was forced out, that we have mayhem, violence, and orgies therein.