“When it becomes necessary to shoot your dog, do it yourself. It’s neither kind nor proper to hire it out.”
99% of the dogs dropped off at pounds are euthanized within hours because the pounds are so overwhelmed.
Dropping it off at the pound does not absolve you.
I had a friend who worked at the local pound because he loved animals. About the only thing he did was bring dogs from the front door to the back room and euthanize them. His girlfriend worked there too. Sadly, it affected her so much she committed suicide by jumping off a bluff.
As you said, if a dog needs to be put down, put it down yourself. Don't make other people do it.