By a similar argument, why should stores check id's when selling alcohol or cigarettes? Raising kids is not their job either.
The answer is because we live in a society. Society is about families, not just adults. Sure, raising kids is primarily the job of the parents, but everyone helps. Sometimes that results in a bit of inconvenience for businesses.
Excluding kids from businesses that are adult-only isn't very kid-friendly, but it's the bare minimum when there are children around.
The issue is that currently, adding restrictions to what minors can do is expensive, both economically, and politically. It requires the cooperation of a lot of non-government appointed people, and many of them could (locally) sabotage the restrictions.
This limits the restrictions to those with incredibly broad support. Keeping a lot of agency with families on how to raise their children.
Digital age verification, if implemented well, is easy to enact, and hard to sabotage without being noticed. That enables restrictions that 49% of people disagree with. Heck, it enables restrictions that 49% of Congress disagrees with. That could be 60% of people disagreeing.
I don't think it's all that different, because nothing is foolproof. People will still circumvent age restrictions by sharing devices. (For example, borrowing a phone.)
And educating the kids should be, and always has been a collective effort. Even in pre-industrial societies it was true. In the modern world it should be doubly true. I think most people would agree that we need public schools, even though some of them disagree with how sex ed and evolution were taught there.
It’s not the material cigarettes or alcohol that are the problem, it’s lying to get them. Same is true for sins like gambling and explicit romance novels.
The enforcement here is quite twisted: it attracts greedy litigants. Lying is bad, but greed is a mortal sin.
The answer is because we live in a society. Society is about families, not just adults. Sure, raising kids is primarily the job of the parents, but everyone helps. Sometimes that results in a bit of inconvenience for businesses.
Excluding kids from businesses that are adult-only isn't very kid-friendly, but it's the bare minimum when there are children around.