between the two you named:
.223 or .22-250? (your thoughts)
Due to age and probably the urge to hunt other animals as he grows up. I'd choose the .22-250 between your two choices.
A .243 or .30-30 would give better penetration, but the .22-250 is a stone cold killer. I have buddies who live in Arizona, and their guides carry .22-250 for Elk hunts. A few of my buddies use .22-250 to kill large Elk cleanly.
I personally have experience with a .22-250 and whitetail deer up to 200 lbs. It is a very impressive round, and one that I'd like to own myself. I just cannot justify the purchase because I own rifles in .223 and I have a .25-06. The .25-06 really knocks the .22-250 out of the game when it comes to power, power retained at distance, and with the ability to go from 85 grains to 120 grains. The .25-06 pretty much has it covered.
Have you thought of a .25-06? With a 100 grain Remington Core-Lokt they recoil is nothing. Less than a .243 in my book. Of course it is all dependent on rifle design and fit as far as recoil goes when the cartridges have similar ballistics.
The only thing about a .25-06, mine has a 22" barrel, is that the report is huge. It jumps more than recoils, and the report is quite heavy.. The shooter definitely feels the boom coming from that long action speed demon. This adds to "perceived" recoil.
For a 12 year old boy I'd be looking for something in a short action. Which you are. I'd skip the .223. Just because a 30-30 would be a better choice despite felt recoil for a 12 year old when buck fever hits.
If legal to hunt in your area with a .22 centerfire rifle then of the two you mentioned, .223 or .22-250, the .22-250 wins hands down. It is accurate, it keeps the power up down range, and is just devastating with 55 grain soft pointed bullets. Once it gets into the chest cavity of deer or whatever you shoot with it. The organs are turned to soup. I have rarely had to track any medium sized game hit with a .22-250 in the boiler room. The shock and awe of that 55 grain soft point is just an awesome sight to be seen and experience.
If you hand load you can go for 69 grain Sierra Game Kings if your rifle shoots them accurately, but I think you'll be fine with the plain jane .22-250 offerings by any major manufacturer of that round.
With all that being said, I'd opt for a .308 Winchester, and buy a Boyds stock. Have a gunsmith make the rifle fit the boy, then use a nice recoil pad such as the recoil pads that come with the Marlin XL7 or XS7, and as he grows older for $80-$100 upgrade the stock to a full size stock. Or use an aftermarket recoil pad that still left the rifle fitting correctly to the boy's size.
He'd have the rifle of a lifetime. Able to hunt small varmints to large and some dangerous game without worry of "do I have the right gun or enough gun". If the rifle fit him well, and a good recoil pad was used, not to mention the extra clothing for staying warm. Then a .308 Winchester would be the best choice in my book. Especially if you reload, but store ammo would be available anywhere.
So my choices of the two you were thinking again: .22-250
For my son when he gets older as we do not have long distances to shoot will be a 30-30 which I can load down and use 125 grain Sierra FNHP. If not a 30-30 then a .308 Winchester.
Or if he likes long action as every 12 year old into hunting and guns has a preference. I'd opt for a .270 or a .25-06 for him.
That is another good point. Ask him. What does he like? Let the boy shoot some rifles and decide. Then discuss the pros and cons with him. Example. A .223 has less recoil, but doesn't have the potential to knock down a large buck that is on high alert, and a .22-250 would be a better option in that scenario. Also, that a .22-250 may recoil a little more, but you could shoot with confidence at longer distances for example, and at shorter distances he would have a greater percentage of knocking the deer down without tracking if a good shot was taken.
We all have opinions here, and some don't agree. But you should speak to the boy and ask him about what he likes. If he doesn't like it, no matter how much or how little recoil it has. He isn't going to shoot good with it.
Good luck, and let us know what you decided on.