This is just one example, but there are others. One day a woman walked in the door with a long line of sutures below her bottom lip and another semicircle of sutures on the side of her face. Her dog had literally bitten her bottom lip almost completely off and then came back and bit her in the side of the face. The attack was completely unprovoked, she was lying on her couch. The dog had a history of unprovoked aggression ever since they got it at 8wks old (rescued from a drug house), with each attack becoming more severe. She had tried various trainers & meds and the dog didn't improve at all. She also had a child in the house. No rescue was going to take that dog, and she was a liability for anyone to own. It was sad because when she was nice, she was super sweet, but you never knew when she was going to turn. Because of the bite we needed to quarantine/board the dog for 10 days prior to euthanizing and we all saw how she would go from sweet nice normal dog, to vicious in an instant with no provocation.
Like many others have said, it's really not black and white. You have to look at everything case by case. If someone brings in their healthy cat and says they want it euthanized because they don't want it anymore, you have a few options you could offer first; if you are able you can foster it and find it a new home or you can offer to take it to a rescue yourself. If neither of those options are possible or the owner declines them, then you have to think what will happen if you don't euthanize the cat? Maybe they will just keep trying different vets until they find one that will, or maybe they will take the cat home, place it in a garbage bag and beat it to death with a crowbar. People do that kind of stuff and often get away with it. As a vet that can offer that animal humane euthanasia, it's not the wrong thing to do.
I would definitely have a hard time deciding with the latter situation, and I would hope the person would be reasonable enough to consider surrendering a pet to a rescue to give it a chance, but not all people are reasonable. Like this one guy who wanted his fully vaccinate, 100% indoor cat euthanized and tested for rabies because it lightly scratched him on the leg one time. I have connections with a cat rescue and they agreed to quarantine the cat for 10 days and then attempt to adopt it out or possibly be kept by one of the rescue volunteers. The owner refused the offer stating that even though it was only one scratch in 5 years, he felt the cat was a liability and didn't feel right allowing it to be adopted out. The practice owner refused to do the euthanasia and I'm sure this guy just went somewhere else to have it done, but the doctor did what he felt was best in that particular situation.