Ellis style is all about letting the dog decide his actions, and then rewarding it when it makes the choice you want. for instance, to teach a pup to sit, we lure the dog into a sit using a treat. when he sits, you mark the reward with a word (some people use clickers) and then treat. You end up with a pavlovian response, where when the dog hears "yes" or "good" he expects a treat, and gets excited. After awhile the dog will start sitting on his own, and you mark/reward that. Later on you can add in the command, when the dog is performing the action on its own.
The difference between the two methods is that with the ellis style, you end up with a dog that WANTS to work, and looks to you for direction rather than being dead inside.
I could go on more, but don't really have the time.
They used (parts of) Koehler, the slow way. We used a pinch collar and a long lead, and a lot of treats, and went very slowly. In total we spent probably 18 months in obedience classes, went all the way through advanced. We spent at least 30 minutes a day, every day, working on training.
It worked. She started out being the worst behaved dog in the class, to the point that my wife disavowed she knew me when I had the dog. But by the end, she was the best dog in the class and it wasn't close. And she turned out to be the best dog to ever live. She literally saved my daughter's life twice. We could let her out into our unfenced front yard and know, without a doubt, that she would scope out the property lines and alert us if anything crossed them but not go over them herself... unless the kids ran off, in which case she would tail them and bark when my wife called to let her know where they were. It was like having a third adult in the family.
It's been ten years since she passed, at age 15, and I still cry when I think about her.
Done wrong, it isn't Koehler.
There have to be consequences to choosing to blow off a command.
You _can_ definitely use positive only training to teach an alternate behavior (e.g. recall or distance down) and proof it under progressively more difficult scenarios (distance/duration/distractions), backing up every time the dog makes a mistake. Until that command is such a strong habit that the dog will reliably do the behavior on cue. And in the meantime prevent the bad behavior (with a leash) to prevent ingraining the undesired behavior.
Not saying negative reinforcement _should_ not be used (it may be more effective, and may even be the difference between achieving or not achieving the desired behavior with the amount of effort any particular trainer can put in). Just that it's not as absolute as you say.
Of course that takes a lot of training, where you can control the situation to give the dog a fake choice (by leashing him) and/or ensure the bad choice is the right amount less awesome than the treat.
I am not advocating against negative reinforcement here. It can help build good habits by steering the dog towards the good choice and preventing the bad one. I'm just disagreeing with the notion that the end result should be the dog deciding based on expected reward and punishment.
(I know this is quite idealistic.)