Your point about insects and such is accurate, which is why I specified mammals and birds.

Domesticated farm life and animals can be quite hardy when left on its own to breed for more robust characteristics. What you would have in a thousand years won't be a poodle, but it will be a dog. Pigs brought into areas for farming have turned out to do pretty good in the wild.

Some animals might have become too specialized to survive. You have to be able to breed enough generations to toughen up the species. I'd have my doubts about the domestic turkey. A farm/genetics expert would have a better grasp of this than I do.