"Those focusing on the child's age are seeing a tree and not the forest"
Absolute truth. Really, this isn't even a parenting issue, but one of gross negligence. No different from the woman who bump-fired a 500S&W revolver into her head after being given one unprepared for the recoil. A kid younger than 9 can swing an axe or hatchet safely (I did, in Scouts), but not until a lot of practice and demonstration of safety practices and technique is done. Why? Because you can't control a person with an axe; if you are close enough to guide them, you are in the kill zone.
The Uzi isn't quite that bad, in that a person can assist a shooter from behind without placing themselves in danger, but as we saw, it is still difficult for an instructor to control the situation effectively. Even holding her grip on the weapon properly and preventing the muzzle from rising would be at best an incredibly awkward maneuver, and the instructor would still have no ability to end shots being fired until the shooter relaxed their finger or the mag ran out. As was said, the parents ignorantly assumed the scenario was carefully controlled and safe (hard to blame them, I imagine the trained authorities told them exactly that), but in fact it was not. It was only as controlled as the instructor's discipline, which as we all know will lapse, the instructor being human.
The best course of action would have been to better control the situation by either affixing the gun (to control muzzle direction) or restricting the ammunition available until the shooter demonstrated the ability to master these limited scenarios. Yes, it was possible that this would be the kid's only exposure to full auto guns in her entire New Jersian existence, but that doesn't make it an intelligent option, same as flying an airplane or driving a race car (or four-wheeler!!!).
Hopefully this (these) incident(s) give the 'tourist gun range' community a bit of pause; that their entire operation is solely based upon their adhering to safety principles, which only the operators have the knowledge and wherewithal to practice, even though many of their shooters cannot be so controlled all the time. Cost of doing business, I suppose, but it's a might steep one. I suspect this place may well be closing its doors before long, since the criminal negligence/homicide lawsuit practically writes itself (the former a suit by the kid's family, the latter by the instructor's family).
I really hope we don't find out that the instructor felt pressured to help the kid shoot, in order to make a sale...
TCB