I have a Stevens Model 94 .410 that I have owned since I was 8 (20 years). I have killed well in excess of 100 squirrels with it. I have shot squirrels stone dead at 40 yards with it.
However, I have lost a lot of squirrels that crawled in holes and the like. I have also bashed the heads in of many that were trying to crawl away.
I have never patterned my .410, but I don't think it would be even an average turkey gun. I also can't see having the pattern density to kill a turkey much past 15 yards.
Now, the original question here was the use of a .410 to start a kid turkey hunting. Let's think about this for a minute. When youth turkey hunting, you have two things to think about. Number one is recoil. We eliminate the 10 ga and 3.5" 12 ga loads. I don't care if the 13 yo punk says he can handle those 3.5" loads. Let's leave them at home.
The second thing to consider is the ability of the youth hunter to quickly acquire the target and make an accurate shot. This is probably a bigger consideration than recoil. Target practice will help but a lot of this comes with experience. Considering that youth hunters will by and large be slower to get on target and less accurate with the shot, we want a gun with plenty of shot and a pattern that is dense but not too dense.
The .410 makes the cut on recoil but fails on the second consideration. Most gobblers will not walk-in to 15 yards and stand motionless, while the youth hunter steadies himself and makes a perfect shot.
I still squirrel hunt with my .410, but I pass on as many shots as I take, and limit myself to high percentage shots no further than 20 yards. The best youth turkey gun is a 20 gauge. If he/she is not big enough to handle the recoil of a 20 gauge, then they probably just need to tag along on a turkey hunt and save their shooting days for later.