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 the responsibility of websites, then parents would still have been given zero controls 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".