Getting the schools to change approach requires political action against powerful lobbies. While a private school needs good outcomes to justify it's prices so gravitate to systems that produce that outcome.
I suppose it doesn't have to be a "monolithic curriculum" that addresses this kind of problem. But whatever it is it better be functionally equivalent in its ability to successfully thwart clever bad faith exploits. Otherwise you get the less populated parts of the U.S. "teaching the controversy" in biology classes as part of your "greater diversity of ideas."
The cost of autonomy is that some teachers will teach creationism, but it also means the people who are closest to & have the biggest stake in the learning of each child have control over how that kid develops, which in general seems better aligned with incentives.