Many air rifles will definitely do the job of humanely dispatching small game and birds and perhaps medium game (under near perfect conditions). However, depending on just an air rifle for a survival situation would be as short-sighted as planning and preparing only one means of starting a fire.

The supplies needed to set traps and snares for small game, birds and even large game are less expensive, lighter in weight, store more compactly and are much more productive than alternatives (firearms and bows) in almost any survival situation lasting more than a few days unless you are forced to stay on the move.

Air rifles are a viable addition to add depth to a survival battery and for inexpensive practice, pest eradication and recreation but their limitations eliminate them as a viable choice for a primary survival hunting weapon IMO.