Eric,
Trying to keep focused on your stated need for your son, here’s what I can tell you about chamberings and rifles that I actually own and have experience with.
Tegg brings up a good point. Is now really the right time to bring in a rifle? Is a shotgun and slugs an option even if you CAN had your son a rifle, the question is should you? Not saying yes or no, but I think it’s worth considering if an auto-loading 20 gauge would be possible starting point if recoil is really a major issue?
For rifles, far and away the best choice I can see for your needs and your son’s build would be the .243. This is a very capable cartridge with relatively mild recoil.
Overall, I think there are a lot of people out there who are quick to disregard this cartridge without actually having any experience with it. 4 years ago, my father found himself using my .243 when his .30-06 went down with mechanical issues. He was absolutely stunned when he dropped a buck stone dead in its tracks at just over 100 yards with a 90 grain Nosler to the boiler room. The deer literally fell over sideways without taking a single step after impact. The .243 hits them fast and hitting fast with modern bullets means they get hit HARD as well. That combination is much more of winner than most people give the .243 credit for.
The .22-250 is not a fit. I’ll just say it flat out. On a practical level–and I own both a .22-250 and a .243–there is almost no difference in the recoil between these two cartridges in equal rifles. The .243 kicks only a miniscule amount harder. The .22-250 is the King of the Varmint Rifles for good reason, but things get wacky when you try to force feed it deer-capable loads. A young hunter doesn’t need the baggage that comes with creating a situation where you’re trying to force a varmint rifle to do double duty.
There are a lot of “tweener” choices that sit between the .243 and larger, harder-kicking chamberings. 7-07, various .25 bores, 6 and 6.5 MM options, .270, .308 etc. To me, it’s not worth talking a lot about these because all of them are a significant step up in recoil.
Grouse