Often the no-kill shelters are the ones who do the full vetting.
Simply just correcting you.
There are no-kill shelters that allow their animals to suffer, and there are “kill” shelters that do not. I’ve observed that when you tell someone you work for a “kill” shelter, they assume that you put-down animals that cannot find a home. So you must correct them and say, “no, only the ones who are in pain.”