My vet has said that is it very safe for birds (I don't have small ones, just Macaws) and he is VERY experienced with the use of it and I think that might be a factor. I would have more trust if the vet is an Avian vet, not mainly a dog and cat vet.
Either way, if the leg has gangrene, there are not many options left and if the little one does not make it because of the anesthetic, that would be a more humane way to go vs leaving the gangrene as that would be a slower death.
Hoping for you that your canary makes it through, please keep us updated.