I don't know for sure. I started shooting at age 6. I didn't kill a deer until I was in my thirties :x
I have been after small game almost as long as I have been shooting and I rarely had any supervision. Not saying it should be that way, but that is the way it was. They also tried to teach me to clean fish earlier than that, but it took me a couple years to do it on my own and do it right.
So 99% of what i know of hunting and fishing came from personal experience, Field and Stream, and Outdoor Life
My grandma always made sure I had a subscription to field and stream. Believe it or not that kept fish in the freezer and kept me out of trouble. So very smart of her to do that. Grown ups worked all the time, Grandma worked until well into her 60's.
I think it's up to you and you gotta use your best judgement on your kids and their maturity level. My girls wants to go hunting with me, I think that itself is a blessing and not something you should piddle around about. As soon as they can place the shot with an adequate caliber for deer, if they have been exposed to dead game and understand you have to take a animal's life and still show desire to hunt then they are ready. That doesn't mean you gotta scare them with talk about death and killing. You will know they are ready by the eay they talk about it.
You have to feed their desire to hunt and fish or they will lose it in their adolescence and may never get it back. So if you want them to be your hunting partners and your grandkids to do it too, you gotta feed that desire and make those good memories.
My dilemma with them isn't a lack of maturity, but they are very small. My ten year old is ready mentally, and her marksmanship skills are good enough, but not quite ready to shoot a powerful rifle because of her physical stature.
My boys are actually my stepsons. Their dad had them in the deer woods very young and they could not shoot any centerfire caliber comfortably, but the desire to get a deer was strong enough that they could handle it just fine. A lot of grown men hunt with rifles they can't shoot comfortably so there is that.. So my boys have a whole lot more experience deer hunting than me. But we're still working on their ability to hunt in general. Their dad put them on deer and made it happen for them before the skills were learned, but I think that is OK. Might be a little bit backwards to some of ya'll but what is important is they have fun times and good memories and they want to spend time with you and learn. That is what you want.
Kids typically won't learn to hunt before they make their first kill I think, because they are not doing it necessarily to put meat on the table. The meat is a bonus of the whole experience. You're not teaching them to hunt to survive, you're sharing it with them and teaching them a way of life, self sufficiency, and responsibility.
Just expect to cut your time in a stand short, they don't have the perseverance or attention span to sit in a stand all day or the perseverance and endurance to spot and stalk all day. You gotta make it fun for them and that means sucking it up and cutting it short sometimes. That has been my experience fishing with them, unless they are really biting good you have an hour or two hours tops before they lose interest. Learning patience comes when the desire to get a big fish outweighs the desire to go home on a slow day. Hunting I think will be a similar experience- maybe I ought to get them on doves or squirrels first on that note.
If your kid has the patience and desire already, good for you.
Me personally, if my daughter could put three consecutive shots on a paper plate with any rifle I already had, we'd have been in a deer stand somewhere this year. I just got her on a rimfire this year and I have no doubt she will bag some squirrels. So whatever opportunity comes up next to hunt I am getting her in the woods, probably with a .410 and a red dot instead of a .22 to go after squirrels so she can learn to manage the recoil a little. She's ten.
Starting them young is smarter in my opinion. They just need constant supervision. You're not turning them loose on their own with a rifle to put meat on the table, hopefully at least. In NC they have to be 17 to hunt alone anyway. By then they should be damn good and ready.