Just from my own personal experience: one of my own birds has once got out (through the cat flap that was taped open for our useless cat), but only made it as far as the patio before deciding it was far too scary out there and flying back in when the door was opened. Another of mine was hand-raised in captivity, ringed and released. 2 weeks later he landed on a lady in the South of England and happily stayed on her shoulder as she went into the house.
A friend of mine has a jackdaw who comes and goes as he pleases, staying out for a few hours at a time, or (rarely) overnight, but generally comes back before dark to spend the night happily in his cage.
Granted, these examples are imprinted birds, but if captivity is all an animal has ever known, even one that's not imprited, given sufficient space to exercise, a stimulating and enriching environment and suitable quantity and variety of food will be perfectly content in captivity.
So no, I don't think an animal that's been born and raised in captivity necessarily yearns for freedom. A lot of humans see a cage as a prison. Certainly my birds see it as home, their territory, and I'm sure other captive animals are the same. Freedom means roaming without limits, but also not knowing where your next meal's coming from, whether you'll be able to find warmth and shelter in bad weather, or whether something's about to try to eat you.