Which is kinder? To take the dog somewhere it loves to go, a familiar setting where it feels comfortable because it has always been happy there and then quickly kill it with a bullet in the right spot, or to take it to a clinic where it is frightened by all the smells and the memories of scary experiences, placed into the hands of strangers and set on a cold hard table. Most dogs don't like being on tables by the way, it's very frightening for them. Then the strangers use clippers to shave part of it's leg, also scary for a dog that has never needed clippers before. Then a long needle is jammed into their leg, into a vein; I've had surgery a few times, this has never been a comfortable or pleasant experience. Next comes the nasty drug that at first will give the dog a feeling of being out of control before it succumbs to it and dies in the hands of people it has probably never known. After that the vet will give the owner a huge bill that the average person can barely afford.
So in the end which is kinder? To be happy and then just gone? Or to be given over to strangers in a place that you have never been happy while sick and in pain and have them do scary and potentially painful things that you don't understand while unable to ask your person to please take you home. The man who shoots their sick pet (provided they know how to do it right) that they have loved all their life isn't cruel. The person who arrests them for doing it is.