I'm not opposed to the idea of a sling but that particular one doesn't look the least bit comfortable, the back edge looks like it digs into the stomach and the dog's butt is hanging. I feel like they could slip out the back pretty easily.
I know with some of the worst fighters for nails, if you can just hold them up in the air in such a way that they can't push their feet against anything, they'll stop fighting you (small dogs of course). I assume this works for that reason, the dog knows it can't run, the dog has nothing to push/fight against, so the dog gives up.
Obviously I'd prefer counter conditioning, but for times when something just has to get done and there isn't time to train first, you try to do whatever will work with the least stress and danger. While the dog is still going to be stressed by the procedure they'll probably feel better in the sling versus being pinned to the floor by someone or wrestling with someone. I feel like when you are trying to physically restrain a dog that fights they can start to panic, whereas when they physically can't fight, as in the sling, they don't freak out so much.