Dogger. You make some interesting points when considering the .260 Rem. and 7-08 Rem. However, and I admit to great prejudice, I would still go with the .308 Win. First off, I have several .308's with barrels ranging from 18 inches to 22 inches. Velocity from My Ruger International with 18 inch barrel is 2,550 FPS with a 165 gr. bullet. Recoil is not that bad. BTW. That rifle weighs just about the same as your Marlin, and is one of my all time favorites. My wife tried it out, and to keep mine, I had to buy her one of her own. And she HATES recoil. My next suggestion, is let your boy use the 30-30, and you use the .308.
One of the problems I see with the .260 and 7-08, is that not all the ammo makers load for it. Everybody does the .308. Might be a problem when out in the rural areas.
Everybody has/sells 30-30 and 30-06. I would think that the .308 is probably now in that same category.
Winchester makes a push feed Model 70 called the Youth/Ranger in .308 and .243. The stock is sized to fit smaller children and women. They aren't bad. I won one in a raffle, but had to restock it with a Ramline because it was too short for me. I've shot groups as small as 3/8's inch with it.
I see you do not reload. Surely there is a gun shop in your area that will make up loads for you.
BTW Art. refering to my comments on 220 gr. bullets in the .308. I have only experimented with the Sierra round nose bullets so far with excellent results. Seems that Hornady's 220's are scarcer than hens teeth around here. I never planned to use a spitzer boattail on the .308 though. They're too darn long. I was seeing if a short range brush load for elk might be possible. As I grow older, the heavier rifles get less attractive when it come to hunting the high country. BTW. The figures I quoted were from a 22 inch barrel. Haven't gotten around to trying them in shorter tubes.
Paul B.
...Drill the bolt handle, turn the magazine into a sieve, and use an aluminum floor-plate...Well, it's a start.