When you click on a search result, you load a new page on a different website. The new page would once again come with a header indicating the content rating. This header would be attached to all pages by law. It would be sent every time you load any page.
Assuming that the actual problem here is the difficulty of implementing reliable content filtering (ala parental controls) then the minimally invasive solution is to institute an open standard that enables any piece of software to easily implement the desired functionality. You can then further pass legislation requiring (for example) that certain classes of website (ex social media) include an indication of this as part of the header.
Concretely, an example header might look like "X-Content-Filter: 13,social-media". If it were legally mandated that all websites send such it would become trivially easy to implement filtering on device since you could simply block any site that failed to send it.
> A lot easier to know things are working if ...
Which is followed by wanting an attested OS (to make sure the value is reliably reported), followed by a process for a third party to verify a government issued ID (since the user might have lied), followed by ...
It's entirely the wrong mentality. It isn't necessary for solving the actual problem, it mandates the leaking of personal data, and it opens an entire can of worms regarding verification of reported fact.