The issues HOW they are put down.
For some reason, people are less squeamish about spending a day taking a dog to a veterinary clinic and having them put an IV Catheter in and injecting pink juice than they are about even thinking of putting a bullet in a dog's head. The bullet is probably much more humane as it is quicker than a trip to the vet, which most dogs hate.
Yes, and people don’t understand backwoods way of life. When we lived in the mountains, a neighbor had a dog that he let run loose. It was a large aggressive dog and it started extending its “territory” into our land. It began to stalk my husband. One day it cornered him under our carport and growled menacingly. It did not attack, my husband went inside, but called the neighbor and told him the dog was becoming threatening toward him. The neighbor apologized and shot the dog. We were shocked, we just wanted him to chain the dog up or something.
But here’s the thing, we had bears and coyotes around, and people would keep large aggressive dogs loose on their land for that reason, to discourage wildlife and vermin, but the dogs had to be people friendly. And these were poor people. The idea of spending money to take a dog to the vet to be put down would be crazy to them. And like you said, they knew it was more humane anyway. To these people a dog isn’t a family member, it’s a tool for survival. Hunting dogs, guard dogs, working dogs. If the dog wasn’t a good fit for the job, you put it down. That’s the culture, that’s how they were raised, how their daddy did it and their granddaddy did it.