No, the browsers would need to reject the sites unconditionally since no one is going to enable parental controls if it breaks everything. Otherwise I expect the current situation of parental controls not working and thus everyone avoiding them and complaining would continue.

Recall that this is exactly what happened with TLS. When browsers started gating all new features behind TLS being active suddenly all the mainstream sites had it working across the board in record time.

The first step is to get Google and Apple to set a date after which adoption is mandated. Provide an easy out for site operators, such as placing a text file at "/.well-known/content-rating" with "tag:all ages" inside to opt the entire site out rather than sending a header per resource or tagging html elements or whatever.

The second step is to approach legislators with this standard and a now very high compliance rate in hand and suggest that they enact a law requiring that such ratings are accurate for certain specific categories (presumably porn, gambling, social media, and user generated content).

The third step is getting governments to do spot enforcement often enough to prevent the system from falling apart.