You can still get hardware that you can install your own OS on. But you have to be deliberate about picking it out before a purchase, rather than hoping to unlock a random carrier phone down the line. For example my phone is a Pixel running Graphene. It has a locked down bootloader that could only be unlocked with the online consent of Google. While this most certainly chafes me (and if I could snap my fingers and make such schemes blink out of existence I would), I do have to admit that it really isn't that debilitating.

The unlocking process zaps the userdata partition. This security model would totally suffice for locking down a child's phone. If the child zaps their phone and erases everything on it, then the parent can handle that out of band.

For the general problem, I would say that there has been a longstanding market failure here, in that parental control software isn't widespread or straightforwardly usable across different websites. Your 3 points don't really address that. (2) has been doable on standard desktops forever, and (3) just pushes mobile devices back towards the capability of desktops (which on its own is laudable!). But standard desktops have had these capabilities for decades and still haven't evolved the kind of straightforward parental controls that most parents are demanding.

I don't think it's a market failure. The reality that password-gating software installation at the OS level can be done on most desktops but not most phones is the opposite sign of a market failure. Mobile OSes have increasingly stripped down capabilities in recent years precisely because of anti-competitive practices. The reason standard desktops have not evolved even better parental control features is not because they're not doing better than phones under a free market. They are already doing better in spite of the fact that most kids use desktops a lot less than they use phones. It's just that the absolute level of demand for parental control features has been low until recent years, and even this recent wave of demand is somewhat manufactured.