Once you have a content label header it's pretty easy to build more complicated systems on top of that. Like the site could send "compliant with x regulation" in the content labeling header, and then that would tell the kid's browser it doesn't have to completely block the page but can instead reload the page with a request header saying "filter out a, b, and c content" and expect that to be complied with.

That could be a fingerprinting vector though so maybe depending on privacy settings it just blocks the page. Really these are all solvable problems that web standards orgs have dealt with before; you just need to create the incentive for such systems to exist.