Kids are fine training with proper instruction, with full range of motion exercise, with correct coaching. It's not any worse than it is to have them doing gymnastics when they're four or five, but you have to teach them the correct movements.
That's mythology. Find me the examples of epiphyseal damage done in barbell training done correctly. There aren't any. It's pediatrician bullshit, and it essentially boils down to people speaking outside their specialty.
Pediatricians, doctors, and orthopedic surgeons are trained in medicine, in how to repair things that are already damaged. That doesn't mean they know how to exercise. Just because your mechanic knows how to work on your car doesn't mean he knows how to get you around downtown Boston. You don't teach little kids how to do these exercises incorrectly and then come back and say that the problems caused by doing them incorrectly are the fault of barbell training.
Furthermore, kids aren't strong enough to use enough weight to hurt themselves, which is one real good argument against quarter squats. You can lift big weights in quarter squats. But if you have to go all the way down below parallel and stand all the way back up with the weight, you're not going to be able to do it until you get really strong. That goes for adults and children.
I have a bar in my gym that weighs 11 pounds. I can make that bar weigh 12 pounds next time. And then I can make it weigh 13 pounds and 80 pounds and finally 245 pounds. I can make my barbell directly scalable to the ability of an eight-year old kid.
Conversely, you put that same eight-year old, 75-pound kid on a soccer field and have him run head-on into another 75-pound kid... is that event scalable? Uh, no. "Hey, undeveloped epiphyses. Still open epiphyses!" And then put cleats on him, have him spin and kick and get hit in the knees.
Yet, how many orthopedic surgeons and pediatricians will recommend against kids playing soccer, versus recommending against kids participating in barbell training programs? Doesn't make any sense, does it? But that's the conventional wisdom. Conventional wisdom is that soccer, which is in fact the most dangerous sport on earth, is fine for kids, but barbell training will stunt their growth. Stupid shit like that.