I agree that parents should have much more responsibility on these matters, but not in exchange for state influence, at least in the US. 40M+ adults lack basic literacy to inform themselves well enough to pass on to their kids. Another 65M only have enough literacy to make low-level inferences and compare/contrast [0].
I’d rather govt (which has many channels for advocacy even if flawed) influence choices than corporations completely control the panopticon of choices (where small time spenders have very little consumer power).
But we’re not far apart. As you mention schools (of the state) are a good place to start.