There should be a children’s Internet, just like there are children’s libraries, and child-locked devices should give access to it. Adults should be able to use the children’s Internet to see what’s there and children should be able to use the adult Internet when supervised by parents and teachers.
Technically, the only thing cooperating websites need is an http header indicating that the client is a child-locked device. Websites can disallow creating accounts or logging in from child-locked devices when they’re only appropriate for adults. There can be laws prohibiting advertising on the children’s Internet, etc, and legit websites will have to follow them. At no point does a website need to know a child’s age or anything else about them. Vendors selling devices are responsible for not selling unrestricted devices to children, but that’s easier than making every website do it.
Since the Internet is still a dangerous place, for non-cooperating websites, child-locked devices do still need the usual whitelists and/or blacklists.