Well who am I to ignore a gun-related thread... =)
Everyone (and not just here, but the media and public in general) is hung up on the type of gun and the age of the shooter. Neither of which are the root cause of this mess.
Children shoot guns all the time. The environment (of which the adult plays the biggest role) is the real issue in this. (This could just as easily have been a family renting some quads, getting absolute crap instruction/training, and having none of their own, and a child zooming off a cliff. The materials aren't the issue.)
As already mentioned by a few above, the 'fun factory' aspect had something to do with it. Although I don't consider it to be the largest issue, as even that COULD still be done safely. Scottsdale Gun Club has 'machinegun adventures' and has managed to pull it off the right way. The business obviously didn't place as much emphasis on the training of either the child or the 'instructor' as they needed to, but seemed to be more in a hurry to get to the fun part, get the tourist's money, and move on.
The real issues from all the information out there so far lie in the deceased. First and foremost, a complete lack of any training given to the child. This is obviously not a 'class' or anything meant to make the shooter proficient, but even sky divers and those renting quads or whatnot get some basic safety information from any reputable vendor.
Next up was poor process. I've been introduced to shooting twice in my life (once at age 12 with a shotgun at a VFW, and again by a family friend at a local shooting range with a 9mm handgun.) In both of those cases, and in every case I've ever personally witnessed or been a part of, one of the BIGGEST rules is not handing a novice a fully loaded weapon. One round. That's it. The issue is more pronounced with something like an Uzi, but the same issue is present with any semi-auto or full auto gun... recoil will redirect the muzzle, and an inexperienced shooter is fairly likely to slap the trigger again as a result of that recoil. If there isn't another round in there, nothing happens.
When I shot a full auto H&K MP5 for the first time the above was key to the experience. One round was in there, and there were a lot of mag changes and reloading until the basic muzzle flip was understood. Move on up to the 3 round burst to get a SMALL idea of what full auto does. Once that was under control, then and only then was I given full mags and let loose, and even then it was still a requirement to shoot in short bursts first. (Suffice it to say I only WISH had had the tens of thousands of dollars it would take to acquire one of those for myself, as it was an AMAZINGLY fun experience. Alas, mortgage/vehicle/food/etc. all take the fiscal priority... for now...)
The same process (or very similar) should've been used in the case of this child, but instead it was all about getting them in and out with some gun pics/video for the grandparents/facebook.
I have no doubt the deceased ran things the way he did simply because that's what the 'job' entailed. His background suggests that he certainly should know better the challenge of controlling something like an Uzi. That said, even if the way things were run were by the business' policy, he should never have followed that. It was wrong, and we all see how following stupid orders/policy can end. In the end, the fault is ENTIRELY on the instructors shoulders, and unfortunately he paid the ultimate price for it.