Sometimes the most humane thing is a quick end.
Call me heartless, but it's the truth.
As much as I hate it, I agree that sometimes the injuries are severe enough that it's more humane to end the suffering of the animal. Having said that, I wouldn't be able to do it personally, and even knowing that it's the right thing to do, I'd cry like a fool once it was done (by someone else!)
In this case, however, I feel there's enough of a chance that it should have been brought immediately to a vet or a rehab center. Sometimes it's not clear and someone who is not familiar with birds might be killing it prematurely.
And for crying out loud, stop throwing the poor thing into the air! Bring it to someone who is qualified to make the decision, and if the bird is injured beyond hope of recovery, then it can be put to sleep less violently than with a pellet gun.