I don't think that there is a need for a technical solution to that, though. In the Kagi example, probably they trust that their users won't do that, and someone could already resell searches this way (e.g. write some kind of proxy). Similarly, an adult can already help a kid get access to stuff they shouldn't. But the point is to make it harder for kids to do it on their own, for their own sake.

It's not computer security, where your system is "as weak as the weakest part". We don't care if a few kids access social media: the goal would be to make it such that the norm, for kids, is to not have social media.