The problem with your proposal is that it's already the status quo. Various whitelists exist but service operators don't generally bother with these things. What you end up with is a largely unusable experience if you enable whitelist filtering.

The core problem is the lack of buy in. Unfortunately that likely needs to be forced. I think it's not unreasonable to legally require people to make a claim about the nature of what they are serving up. They already need to be aware of the legal status of what they're doing anyway so it hardly seems as though making such a determination should pose a burden when you consider that it's an alternative to either requiring ID, requiring the client send age bracket information, or other heavy handed interventions. The choice here isn't "the status quo vs a header" but rather "some other age related regulation vs a header".

An easy way to enforce this "voluntarily" (ie coerce) without sending government agents after every small time website operator would be to require that mainstream browsers and other client software (based on MAU or similar metrics) refuse to process content that does not send a classification header. Doesn't matter what the header says or what the status of the user account or parental controls or whatever else is, it has to send the header regardless or it will be blocked without exception. That would presumably trigger broad compliance with the relevant regulations.