I was 9 when Dad gave me a sporterized 6.5 Swede. At 14, I swapped it for a guitar and Dad gave me a sporterized Persian Carbine 8x57mm. 47 years later, the Persian Carbine is still my go-to rifle for all occasions; mice to moose. Your kid is 12. You've taught him and brought him along. He has learned to trust you. ANY rifle that doesn't fit him is going to be uncomfortable to shoot. I've had people say my 8x57mm kicks like their 338Win Mag. To me, it doesn't kick at all, but the stock was made for me and it fits. IF you do not handload, you are short-changing both of you. Handloading, you can virtually make any cartridge suitable for him at 12 and let him grow into it.
I'm not a fan of either the 308 or the 30-06, but a wise man once said you can never go wrong with the 30-06, and I'm smart enough not to belittle a wise man. When your youngster is grown up, 30-06 ammo will still be available most anywhere, until he realizes the blessings of handloading. Then, he will create it at will. Really, no wrong choices.
Jeff
NRA Life