When I started showing minis I thought the same thing - why don't they use the quarter system? I learned halves in 4-H and showing arabians, then the change to quarters took place about the time I was done in youth and started showing open and judging. Personally, I prefer the quarter system with big horses and halves with minis. While safety may have been an initial reason for the quarter system, just as much of the reason was that to show halves with a big horse you had to go through all kinds of contortions to be able to see the judge at all times as he inspected the horse while maintaining a position where you didn't block the judges view of the horse. It was a horrible way to show.
With the large horses, the quarter system allows handler to see both horse and judge at all times, and still be in a position of control of the horse, and not block the judge's view of the horse. The horse is larger, the judge takes longer to inspect, and the movements then don't seem "continuous" - unless you have a REALLY fast moving judge. However, when I watch minis doing the quarter system, the handler inevitably ends up in almost constant motion, unless the judge consciously slows down their inspection. The minis don't have to do quarters to be able to see the horse and judge at the same time, and maintain a position of control while staying out of the judge's line of sight. They can see over their horse, so being opposite the judge when he's in back of the horse is not an issue.
The safety aspect may have been an aspect originally, but that was also when showmanship was really presented as a class that was meant to be "how well you can show a halter horse". In most breeds, showmanship is now a class of it's own, that showcases the partnership of horse / handler on the ground, and is only remotely linked to ability to show a horse in open halter. The exceptions to that seem to be generally the pleasure type breeds, who present their showmanship classes much more similarly to the halter classes. So I'm not sure the safety aspect has a place in the class as much anymore. The horses are not shown as "hot" halter horses, where it's more likely they will have an issue that the handler needs to deal with in moving them away from the judge.
Just my opinion. I and the youth that show my horses both know both ways. In 4-H the youth show quarter system. In open shows, if we have a judge who I know also judges AMHR, I have them show halves. If not, then they show quarters. Generally, it's not a problem.
It just seems that most have moved the other way. But we sure don't NEED to be like the rest 
