This is the first I've ever heard of a kill shelter being picky. They're usually the cheapest place to adopt from, and the last ones to care where the dogs go. When I was trying to adopt a puppy from a city kill shelter I was denied, but not by the reasons you would normally hear. I was denied because they had purebred puppies and they were going to be auctioned off to the highest bidder.
OP, from your last response it sounds like your in the military, is that right? You said not to judge all military by what some do, aren't you judging or putting together all rescues by what some do? It's been my experience that each rescue is different. Each shelter is different. Each adoption is different.