All the talk of "shooting until it's down" goes hand in hand with "put the bullet in the right place", although it seems contridictory.
Think about it. We have heard/seen about animals that are fataly hit that still move anywhere from a few yards to several hundred yards. But they are still dead, they just haven't fallen over yet. Those that feel they have "put the bullet where it counts" perhaps see no reason to fire additional meat-ruining, potential for gut-shot rounds into an already dead animal.
Sure, sometimes if an animal is still moving after a shot in the heart/ lungs, a second or third shot may be warranted to anchor it by breaking a bone, if you were concerned about it getting into some spot that would make it hard to retreive, such as a lake. But if you were confident you hit that nice buck exactly where you wanted in the middle of a wide open field on the first shot, do you need to empty the mag into it just because it didn't fall over immediately?
That, I believe, is part of the message being put forth by the "put the bullet where it counts crowd. Once the confidence is built up by practice and experience, an individual can start to get a feel for whether his first shot is good, or if more rounds are needed. And perhaps can wait for the animal to fall over.