I'm a fan of the .243, I started a young boy out with one, he loved it as he was a bit recoil shy. He took three very respectable mule deer bucks, a Texas Dall ram and two pronghorn antelope with that rifle.
remington model 7 20 inch barrel with a youth stock-you can get a full size stock later-it will drop in to real easy! to model 7 with the 20 inch barrel may be a bit heavy for him but the extra weight will help soak up recoil and be easier to hold steady on a rest. .223 will do the job but a 80 grn .243 would be a bit more effective. I have taken several kids under 10yrs old and this set up is just about right for all of them...
Howa is bringing out a 6.5 Grendel in a bolt action this Summer, less recoil than a 243 and similar effective range. Not that there is anything wrong with a 243.
On the Rossi be sure the forearm is screwed on the same any time it is removed. Screw tension on it affects point of impact and accuracy of those little rifles. Or at least the ones I have delt with it did.
My kiddo (6) shot two deer and a turkey this year with a little break over 223. Looked at all the guns and shot a few but this one was perfect for him. He used the fed fusion ammo and they ran about 30 yards. Had a deer pic on my phone to remind him where to shoot and made him point at it before every shot. I think trying to explain where to hit them was the hardest part. Good luck!