In the wilderness-lost scenario, taking a deer with a .22lr might be possible with properly placed head shot at close range, but, not necessary the best way to feed yourself. Squirrel and rabbit would probably be easier all way around. More plentiful per square miles, easier to skin and cook.
If I was hungry I certainly would take the shot at the deer, but it would be a Hail Mary aiming-for-the-head-and-praying shot.
If it were long-term wilderness survival, then the skins of larger game become much more important for shelter and clothing. Then it may call for at least a small centerfire like a .223.
Side note: I do not own either of these rifles, but, I thought I would mention the utility of an M1 carbine or an SKS in 7.62x39. being Carbines, they are lighter than full size rifles, it is easier/lighter to carry their ammo as compared to a 30.06. They are compact making them good in thicker vegetation, but have enough power to take down deer all day long. if they lack anywhere, it would be in the accuracy department, but as long a you weren't trying to make deer-killing shots at 300 yards with iron sights, they would certainly fit the sitaution, lgiht, compact, but still pack a punch for defense and larger game taking.