A ZKP will work as a base, but the proof mechanism will have to be combined with anti-user measures like device attestation to prevent things like me offering an API to continually sign requests for strangers. You can rate-limit it, or you can add an identifier, both of which makes it not zero knowledge.

Parent's proposal is better in that it would only take away general purpose computing from children rather than from everyone. A sympathetic parent can also allow it anyway, just like how a parent can legally provide a teen with alcohol in most places. As a society we generally consider that parents have a right to decide which things are appropriate for their children.

Honestly I think no measure is and should be perfect. It's completely disproportionate. If there's a will there's a way.