You can sometimes find a wildlife rescue person who will provide assistance, expecially for an injury. Broken legs and wings are not that uncommon for wild birds.
Also, getting treatment for an injury is sometimes easier than getting treatment for a sick bird. Wounds require fairly similar treatment, regardless of the species. Illnesses and medicines can vary greatly.
If all elese fails, show up as a walk-in with your injured pet and plead with them to take care of your beloved pet. Other patients in the waiting room would not take well to a severely injured pet being turned away for no obvious reason; htey will think, "could that happen to ME and my beloved Fluffy?".
I am blest to have a wonderful avian/exotic vet who will see my chickens. He costs much less than the vet the cats and dog go to; unfortunately, Dr Burke is a bit of a drive or I would transfer all my animals to him.