There's nothing contradictory about what I've said. Even at the most basic level - teaching children not to play with firearms - it is the parents' responsibility. Some will teach more than that, others won't and don't have to.
I see no difference in this than I do the schools teaching sex ed or character ed or driver's ed. It is not the role of the government to teach my children about sex, about manners or how to drive. And under these circumstances it is certainly not the role of the government to teach my children how to shoot.
This is the fundamental flaw with your argument. Like proving a negative, you can't teach anyone NOT to do something: only TO do something. You can try to constantly reinforce prohibition, but you can't instruct to prohibit.
With half of all American households containing guns, it is simply irresponsible to consign the children of the non-gun owning homes to total ignorance of the subject.