Blemgo

Though this solution also seems to be very flawed, doesn't it? You basically trust another company to manage your child's smartphone and granting it full access to it. Furthermore, that doesn't stop predators, as they could still arrange meetups with their unknowing victims. And even if it captures text messages, kids would be discouraged to use their phone due to their fear of their parents disproving of their friends or their communication to them. Instead, they'd more likely learn the use of "burner phones" by getting a factory-reset phone and using that one instead.

It's the sort of ham-fisted attempt expected by parents that blame their kids for their mistakes instead of their parenting.

Could you explain how it doesn't undermine your argument?

I think the term you are looking for is "Deep Net", although it originally meant websites that weren't indexed by web searches.