The reason the current system doesn't work when it comes to unfettered internet access is either that parents by and large don't actually agree with the threat assessment of said internet access for kids, or they ignore it because the consequences of doing so are disproportionately small. Personally, my vote is that they just don't think it's that big of a problem.
To me, the ideal solution would be more granular and better parental control configurations, especially on the web. Pair that with preconfigured devices that have "unremovable" parental controls, branded as "kids/youth phones." If parents care about this, they'll buy those phones for their kids and the problem is solved. If they don't, then this isn't something parents want, and we shouldn't really pursue it further.