I know if I ever have kids they will be brought into the hunting world the same way I was. I started dove hunting with my dad at around age 4 or 5 I cant remember exactly. He really didn't hunt big game just the birds mainly. I was the bird dog until 3rd grade when I got my .22 for christmas and then for my birthday in January that same year I got a .20 H&R that had been cut down to fit a youth shooter. I had been throughly educated during those first 4 years of just fetching the birds and I felt privileged just getting to go with them and I had helped clean the birds during this entire learning process. I did not get to hunt until I was 9 and I was within close proximity to my Dad and he was purposely letting them fly by so I could get my chance to shoot, I still vividly remember shooting 2 boxes of shells and only killing 2 birds (blew the head clean off my first one). I then got to start shooting on the duck hunts the next year and it was same story I hit 2 birds out of at least a box of shells. I then got my first real rifle a marlin .30-30 for my 11th birthday. I went on my first deer hunt when I was 12 and it was the first time my dad had been deer hunting in 13 or 14 years and he paid $300 dollars for me and my brother to both shoot our first deer. He did not buy a tag for himself because his main goal was watching both of us kill our first deer. The rancher was so impressed he let my Dad kill a turkey free of charge. And I still vividly remember that hunt. I had buck fever so bad and was shaking so bad it took me 5 shots before he finally hit the dirt. Clean misses the first 2 hit him in the lungs on the 3rd and the last 2 were just because he wasn't on the ground yet and if my Dad hadn't made me stop I would have emptied that marlin. And when it came cleaning time he did the brunt of the work but I was paying a lot of attention and helping where I could. And he told me and my brother he would do the first one and then after that you were on your own with him giving pointers over your shoulder. This is how my brother and sister started hunting also and all three of us love the outdoors and love to hunt so I think he was doing something right.
So I think if I ever have kids this is how I am going to introduce them into the hunting world slow and steady. Let them bird dog for a few years and then slowly let them start shooting and then gradually move up in caliber and species.
So to answer the OP question I think they should be small game hunting at age 4-5 but should not be pulling the trigger until they hit an age where they appreciate it a little bit more for me it was age 9 for small game, and age 12 for deer. But watching a 4 yr old shoot deer is pretty ridicules to me, but to each his own.