But we already have that and it doesn't work. Sites don't bother to classify themselves. Whitelists are spotty and out of date. Almost nothing interoperates properly. My expectation given the scenario you describe is that business continues as usual with the vast majority of the internet thus being inaccessible when in "child" mode. As a result no one uses child mode and the vicious cycle perpetuates.

A solution would be browsers refusing to load any site that didn't provide such metadata, not just in child mode but in any mode. I suppose either Apple or Google would likely be capable of unilaterally imposing such a requirement as things currently stand. However I also think that's unlikely to happen on its own for various reasons. Thus legally mandating certain basic content classification categories in addition to an extensible standard protocol for communicating such metadata would (I expect) end the cycle. And rather than the government going after individuals the legislation could be structured to place the onus on mainstream browsers, OSes, and other client software to enforce compliance on their behalf.