Flat out, you can't leave it to parents. For example, drivers have total responsibility for their vehicles, yet cars still have all sorts of safety measures that wouldn't be necessary if drivers simply drove perfectly and never crashed. It's an acknowledgement of human failure.

Parenting is the same. Not everyone is good at it, and it's in the best interests of society to raise kids to be as productive as possible even if (and especially when) their parents don't do a good job.

Public education has been a huge net positive for society over the last few hundred years. It used to be that most people couldn't even read; such people in the modern information economy would just be total drags on public resources. Better to have the state educate everyone as much as possible to ensure a talented, capable, intelligent workforce and society.