I own and shoot a Winchester 1894 saddle gun in .44 Magnum. Just as with the handguns, you can shoot .44 Special ammunition in it, if the .44 Magnum is too much for you.

That's why I urge you to read before commenting - my first choice is a .357 Magnum lever gun or single shot, and limit shots to 50 yards. And, yes, I own several of those rifles, too.

If the youth hunter is under 5 feet tall, he is surely going to need a lighter rifle, with a cut down stock, so why not buy a used rifle and chop that up, then download it to 7 ft-lbs of recoil if it is something like a 7mm-08?

I remember when I first came to this web site and was ridiculed for suggesting the 7mm-08 with 120-gr Nosler Ballistic Tips, downloaded to 2,500 fps. Now, the same people are big fans of the 120-gr BT. Funny what a little actual experience will do for people.

Another suggestion, which I don't own, but seems to work for a lot of people:
CZ 527 carbine in 7.62x39mm.