What are you talking about?
This is true across much of America, now:
https://www.usatoday.com/story/news/nation/2015/06/11/can-an-owner-euthanize-own-dog/71051018/
Yet, shelters can kill as many healthy animals as they want.
Years ago, it was fairly common for people to euthanize their own pets, often with a rifle shot taken behind the shed or barn, but the laws and society's view about dogs and other pets have evolved. Michigan's law says a person has to have “just cause” to kill or intentionally injure an animal, but it doesn't define what is just cause.