All of the calibers mentioned will work for the intended purposes.
One thing the .308 Winchester has going for it is the availability of ammunition in a retail environment. Most of the retailers I have visited in my area have seen some stock of .308 Winchester arrive for their shelves. Mostly the Winchester "Deer Season" loading and blue box Federal.
While no one plans to forget their ammo for a hunt, or run short on a hunt, if it happens, .308 Win ammo is a cartridge likely to be found at a retailer. Now, the same can be said for the .270 Win and the .30-06 Gov't, but those rifies tend to be a bit heavier and longer than alot of kids can handle, one of the reasons the short action cartridges are preferred by many for starting off new hunters.
I started my kids with a Remington Model 7 in .308 Winchester. Loaded up 125 gr bullets for practice and reasonable range deer hunting. Loaded or bought 165 grain premium bullet loadings for elk hunting. It worked out well for both of them.
Well done to you for providing a rifle for a deserving youngster.