I don't like the idea of mercy killings, whether it is animals or people. It just seems somehow unethical in a sickening kind of way, more so than just shooting the thing and eating it as might be the case with a cow. It is sickening because it seems to imply that there is some kind of refined edge to the process of snuffing life out.
It's like the difference between a criminal shooting a guy and leaving him dead on the floor without caring, and some sterile prison which is going to give a guy a last meal, and sit there talking to him, then strap him to a chair and zap him to death at a specific time in accordance with paperwork-mandated instructions.
Why the hell do folks say "I had my dog euthanized"? They say it because it somehow soothes their super-sensitive feelings; as if giving it to someone else to kill in a special way behind closed doors makes the act deserving of a different term. Foolishness! Just tell it like it is. "I killed my dog because its life isn't worth paying for, and I'd rather have Starbucks than pay for dog costs. If it comes to life again, I'll fill it full of bullets or smash its skull." Of course, nobody says that... because we want to believe some hogwash about happy places in which there is a soft, appealing term for everything. The people who "euthanize" dogs are also the people who believe that ham is made by happy smiling animated disney character pigs that never die. You "put your animal to sleep"? Liar! YA KILLED IT!
There is no more twisted perversion that the subtle self-tricking which weak-minded people do to themselves through illusion; and it usually works, except for in cases like this when their fragile, weak sensitivities are forced to become fully aware of reality.