Both connections would use the same "kid unfriendly" internet. You could set DNS servers, but each client can use another DNS server. You'd need a completely separate network (and connection) in order for it to be foolproof.

Interesting idea. Would need new laws enacted and government oversight which I'm not sure how people feel about that especially for minors. Maybe there will come a time where all people (in the US or otherwise) have some form of government document that uniquely identifies them (a trusted identity platform not just an ID) though there is and will be lots of pushback. A simple search revealed this: "The implementation of universal government ID systems, particularly digital IDs, is frequently criticized as a key component of a dystopian surveillance state. Critics warn that such systems can lead to mass surveillance, restriction of freedom of movement, and control over personal, financial, and medical lives." So take it for what you will. Can we do this? Yes. Will we is another matter.

a major part of the problem space, is the web, experienced via the browser is not the internet.

escape the browser space, and you will experience the actual internet, a place where no one is supposed to be, a "Darknet" an illegitimate place that is not commercialized, where you have to make your own choices.