Maybe I was naive in this, but I didn't think the term 'no-kill' automatically prevented a shelter from putting an animal to sleep in a medical situation, the same way we sometimes have to do with our beloved companion animals. If the animal is sick and in pain, and there is only realistically the possibility of a slow and agonising death, I believe it's the kindest thing to do. I want to set up a shelter like this when I retire. I do also understand, however, the money side of things - the sad truth is that it would be impossible for all shelters no be no-kill. The thing I'm wrestling with is where to adopt my future kittys from. I want to support no-kill shelters because they are more in line with my ethics, but if I adopt from a kill shelter I may have directly saved an animal's life.
I don't think supporting a kill shelter in that way is like supporting a pet shop or the meat industry. They aren't making any profit from it or exploiting animals, they're probably just doing their best with a crappy situation. I wouldn't consider it against my own ethics to take animals that would be killed from kill shelters, because although technically you're opening up another space for an animal to come in and potentially be killed, it could equally open up a space for an animal to be adopted.