For wilderness survival I would avoid the 5.56mm, incorrectly called .223 (which is related but different) entirely. The cartridge is overkill for small game, a major long-term food source, while it is too light for deer, a good source of meat that lasts as long as it lasts.
The 5.56 possibly creates a higher chamber pressure and is why it's not suggested to use a 5.56 in a rifle labeled/chambered in .223 and why it is perfectly fine to use .223 in a 5.56 chambered barrel. They are technically interchangeable.
Depends on your deer too. But I agree a .22 and a 30-06 would probably prove much more useful.