If it was my kid, he'd be starting his hunting of big game with a .243, a .250 Savage or a .257 Roberts.

There's no reason to use more gun for deer size animals.....and as for deer running off after being shot.....I've seen it more with the .30-06 than any other cartridge.....it's related to placement and certainly not power.

The .243 and similar rifles have plenty of power and the kids will shoot them better due to the lower recoil.