I honestly wouldn't want a small kid shooting either.
Started my son off on a .243 Win but it was an ultralight NEF Handi-rifle that unfortunately wouldn't group at all with any of the 95-100 gr. deer loads we tried. Why they didn't put anymore twist in those rifles is beyond me but the bullets of more than one box we tried literally keyholed the target. So upon recommendation of a clerk at the BPS we tried Hornady 75 gr hollowpoints, which to our amazement all of a sudden produced little cloverleaf groups. Learned the hard way, though, that soft little .243 hollow points aren't made for deer, though, and a snap cap at the bench proved my suspicion that the gun was so light, boy son was starting to develop a flinch.
More I think about it, I'd suggest a 6.5 Creedmoor as long as the gun wasn't way too big for the kid.