They should but assuming it will "just work" is foolish.

If society wants universal capability instead of depending on parent ability and inclination (rather scattershot) the answer is yes they do have a responsibility.

Sure. But on a school day my kid spends around 7 hours at school. And only 3 awake hours with me. And in those 3 hours we also have to fit in breakfast, dinner, chores, and getting them ready for school. So maybe 1 hour of actual time where we can do something.

There are also weekends of course. But ultimately kids spend about as much time in school as they do with a (working) parent.

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.

1. Kids also learn through play and downtime and simply going with the adult to do things.

2. Too much schooling is detrimental to learning.

3. Most parents aren't trained in teaching. If we expect a parent to teach a child formal stuff, perhaps we should train the parents on how to do these things.

There is no accountability for a failure to raise a child well, so no, parents have no responsibility.

It is of course a conversation one could have on if parents should be held responsible.

Schools are the only force in childhood held even remotely accountable for the success of those children entering adulthood. Which is kind of where the desolation of inner city schools come from - the schools are punished for bad parenting, but have no real capacity (especially once punished via funding cuts) to do anything about it.

Should we expect anything from parents, really?

An absurd percentage of people can't properly learn the simple task of driving even after taking classes and passing a state exam. How do you imagine these people to perform in the task of raising a human being, from an objective perspective? The answer is they suck. Terribly.