Cell phone companies like AT&T could offer kid-lines (with filtered Internet access) and Google and Apple could provide kid-modes on their phones that don't allow VPNs or apps to be installed that parents do not approve of.
Maybe there might already be ways to prevent VPNs/apps, but it doesn't seem to be easy and/or publicized.
Don't worry the kid mode is coming on all devices, one thing you wouldn't want kids doing is sideloading applications after all.
I think this is the wrong approach, an example is youtube kids. There seems to be a abundance of inappropriate content for kids on there. These companies don't actually care about you or your kids they care about profit.
Only (hopefully most) parents care about their kids. They have the power to push a solution as a collective so the solution should empower them to choose and not not take power away from them and others (for example adults without kids). The age verification mandated on a government level constitutes to limiting access to content, and in my eyes that is censorship.
Either a kid phone built on AOSP or a kid-focused MDM system, coupled with kid-focused apps, would seem to be sufficient. No need to go to the carriers.
Maybe there might already be ways to prevent VPNs/apps, but it doesn't seem to be easy and/or publicized.