I'm not sure what I feel about depriving teens of general purpose computing devices, either, which is the logical consequence of both the pseudo-ZKP scheme and parent's "underage signal". I believe most of us here learned programming through being able to run arbitrary programs, and that would never have happened if we only had access to locked down devices. And that habit of viewing computers as appliances controlled by other people isn't going to go away on their 18th birthday either.
Overall I think while there is a reasonable argument in favor of age verification for some types of sites, the harms of implementing it would drastically outweigh any benefits that it should not be done.
Sure, I'm sympathetic to that idea. The point is that it leaves such a decision up to parents, putting non-locked-down computers in the same position as any other potentially-harmful thing you might want to keep away from your kids.
Keeping parents in control also lets them make decisions contrary to what the corporate surveillance industry can legally get away with. For example we can easily imagine an equivalent of Facebook jumping through whatever hoops it needs to do to target minors, perhaps outright banned various places but not generally in the US. If age restrictions are going to be the responsibility of websites, then parents will still have been given no additional tools to prevent their kids from becoming addicted to crap like that.
Shooting from the hip about the situation you describe, I'd be tempted to give a kid a locked-down phone with heavy filtering (or perhaps without even a web browser so they can't use Facebook and its ilk), and then an unrestricted desktop computer which carries more "social accountability".