I think of it this way when I decide whether or not to give a bird a chance to pull through. Birds are prey animals; they know it and it shows in their instincts. A wound can be treated and heal, but an unknown illness can't. When a bird is being attacked by the flock and/or goes off alone to huddle in a corner, I know that it is seriously ill. Otherwise, they hide symptoms and you would never know anything is wrong.
When a bird has settled into a spot to wait for the end, I think it is kinder to put them out of their misery. For them, their instincts tell them that the end is coming. Every rustling leaf and flicker of shadow is a predator coming to eat them. They are just too sick and far gone to do anything about it. Definitely not the way I would want ANYONE to spend their remaining time on Earth. Better to give a quick and painless end vs a lingering fear-filled experience.