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You don’t need anything this elaborate.

Set parental controls on set up, pass a single flag to websites and apps, similar to the Global Privacy Control.

No privacy is lost. Control is handed to the device owner, and implementation is technically trivial.

Would it not be trivial to make a webpage which proxies sites but with the headers removed, bypassing the whole thing?

Can't set local proxy because of parental controls, can't setup cloud proxy because of ... Being a kid.

That's essentially the approach California is taking.