I completely agree. Any vet that will tell you to put the bird down without even having bothered to see it is a pretty poor one and obviously has very little care for animals. Heck, my vet went out and bought new tools just to use on my son's guinea pig when it needed it's teeth fixed.
Everything I have ever read/heard says NOT to put neosporin in the eyes, it even says that on the label. The Terramycin ointment is great stuff though! LOL