Might try N find him a 280 Remington and just get him a single stage Lee press at the same time. l Then just let him figure out which bullet weights he likes the best. If he's old enuff to hunt elk , he can certainly learn to reload his own cartridges.
Saying the same as others, but it is the correct response. Lighter bullets and reduced velocity in the .308 will greatly reduce recoil, or shoot the same load and add a brake and heavier stock. Small adjustments in weight and fit can be huge improvements. Just a heavier scope can help. Also, I assume your son is still growing, between ages 11-16, most young men gain the muscle mass needed to tame even the heaviest load out of a light 308.
We are on the fourth page and still have not heard from the op the age/size of his son.
When my grandson was 8 years old he wanted to go deer hunting with me. We bought him a youth model .243 and put a low power optic on it for him. I had him shoot 100 grain federal fusions at an 8 in steel plate that was at 100 yards.
He shot from standing and sitting until he could routinely hit the target. While building his confidence with hitting steel routinely we talked about making ethical shots and that what we hunt, we owe a clean, efficient kill. Opening morning, a buck came by but he didn’t think he could make a good shot so it walked by as we watched. Later this doe came by and he told me he could make the shot while it was milling around. He killed it with one well placed shot. He has went on to kill other deer with his rifle, each with one shot. My point in sharing this is to enjoy the bonding time and that bigger caliber isn’t always the answer, shot placement is. Learning to manage recoil, trigger pull, cheek weld etc. Happy hunting!