This is probably the one topic that bothers me most about raising animals, especially chickens. God has really been so good to me because in over 9 years of having them I have never had to cull even one chick. I have had some that died of course, but they were just dead when I went out to feed, etc. There have been those that I was treating with antibiotics, etc, that died overnight while I was sleeping, or away, and came back to find they had passed. There have been surplus roosters that I sold or gave away. And I do not let animals exist in unnecessary pain for anyone that may wonder. I am so thankful that I haven't been faced with this as it is both gut-wrenching and heart-breaking. I don't believe that I could do it personally, but my husband would do it for me if necessary.
My heart goes out to all that have had to do this. It can not be easy any way route you go.