Hang tough Katherine. Don't listen.
There are times when it just has to be done even though it should not on an everyday basis.
I am having one done right now. She is going to be the spoiled cat of a nursing home. I'd say that is a better life than the alley I saved her from, starving and cold. And for those who say to rehome her somewhere else instead, look at all the adult cats looking for homes. It is easier said than done. Now she will be a joy to many people.
Oh and I might add that I found a conscientious vet who does local freezing in addition, gives pain meds, and who keeps the cat for three days post-surgery for monitoring.