I disagree that it is all about the size of the gun and the size of the kid is pretty relevant here. I have 3 boys, the youngest will be 13 friday. All my boys are/were extra light. My youngest killed a sweet 10pt whitetail last year with my Tikka 25-06, which he will carry for a cow tag he has this year. The 25-06 is one of my rifles and he likes it so I can wait for him to grow. He was 4'11 and 80lbs with his whitey, and is 5'4 and 100lbs now. My other boys were only less than 100lbs until their sophmore year. I started them with .270 adn .280 which was marginal for their size. Caliber was fine for elk of course, but heavier bullets significant impact recoil. I do not recommend brakes for any kids as that sound may be worse than recoil for generating kick. So is extra light guns. I think all .30 cal's are too much if they are 100lbs or less. No elk in this world is too much for a good 120-140 grain bullet launched above 2700fps, placed in the boiler room. I'm not saying for long range, but certainly 200 yards.
For a YOUTH gun, I would recommend 7mm-08 above all others for those under 120lbs. Over then look at .270/.280/.308. Not until they are over 150lbs and have shot a bit, look at 30-06/7mm/any wsm, etc.
Just the opinion of a dad that has been there and loads a shoots quite a bit with his kids.