Just about every animal in North American has been killed with a .22 in the past. I don't think an air rifle can make the same claim. I've taken lots of game animals with a .22 that I would never consider using an air rifle on. If you want to try to live on rats, squirrels, frogs and the occasional rabbit, an airgun might get you by. If you'd like a chance at anything bigger, better make it at least a .22.
Another thing to think about is the relative size of the rifles. I've got 22's that are much smaller than high power air rifles. Compare the AR-7 you were talking about to any of the 1000fps springers. Which would you rather tote around?
In a survival situation you are likely to have to defend yourself. I don't know how much help an air rifle would be for that. You are likely to just upset someone. At least with a .22 you can really **** them off.