I think that one should not underestimate a childs ability to learn and do right, and at the same time not underestimate their ability to make mistakes. They might have 10 range trips in a row with no violations, then make a mistake. Don't baby them and make em wait until their teens, but don't get complacent once they start doing good, mistakes WILL be made. The grey area is responding to mistakes they make. Its a fine line between responding sternly enough to drive the point home, and yet, not so sternly that you turn them off to it by getting them in too deep doo doo for a mistake.
My kids were shooting 22shorts in the Bearcat at 3 unaided, but I was very close. 22 rifles before they could hold them up. But like stated, depends on the kids. By age 6 they were shooting 45ACP somewhat, and 223's from the rest. (I waited on the bigger than 22 stuff until they asked with no prompting, and did not turn them down.) Both of them could hit a 2 pound coffee can at close to 25 yards from a rest with iron sights, with a Mini-14, by age 6.
but everytime they go out to the range I have the worries in the back of my mind,
My wife said this also. My response? Honey, I'm only human, there's 2 kids. When I hand one a hot gun, my attention from the one kid will be obviously decreased to keep focus on the one with the gun. It'll probably be OK, but realistically, you should be there just in case. She started going also. Solution? GO WITH THEM. No one can be expected to watch for the kids safety as much as the momma.
There's no such thing as too much safety with kids & guns. Let the kids hold the gun and trigger it though. When they were real young, I'd let them hold the Bearcat and just surround them with my arms and held them at their forearms.
Demonstrated the power and finality of guns to them by shooting a shaken soda pop for them to watch. Then gave them the destroyed can and told them to "fix it", "make it better", "this is what would happen to your brother if you shot him", "This is not TV, this is REAL", and so forth.
Loads of fun and never had a tradgedy. Patience. (Go with them!)