Your last sentence is the key!
Don't knock it until you've tried it!
I have used just about every method in the book but I trained Rafi without a prong, collar or leash corrections. In fact, I trained him primarily with play! I have also been retraining my partner's 9 yo dog using similar methods and she has come a long way in 2 years.
Generally people who are purely positive use redirection instead of corrections. Check out Pat Miller's books and articles. She is an excellent trainer and her instructions are very clear.
I used to be almost purely positive but have gotten lazy recently. :-( Rafi is super biddable and doesn't need corrections. In fact, it slows down his learning when I get sharp with him.
For some dogs PO will probably work ok although it might it be slower and less reliable in some cases (this last from JEAN DONALDSON herself).
PO training will not work very well, if at all, for some dogs for some behaviors for some trainers. That is FACT.
A softer more submissive dog will often do pretty well with PO training. A hard headed, more independent, more confident dog will often decide that he wants to do what HE wants to do when he wants to do it; and the redirection and/or treat is not nearly as interesting as what he wants to do at the moment. SO then what does the trainer do?
That is, what does the trainer do if the thing the dog is doing, is much more attractive than what you want him to do?
And please don't say, as I have heard MSO ANY times "Get a higher value treat", unless you can tell me a higher value treat than a small piece of raw meat!
Corrections work! And done right also result in a VERY happy working dog as well as a Reliable dog!
And personally in my case with Baron (4+ yo male GSD with a few DA issues) a much more sociable dog!
Even Jean Donaldson in one of her books says that PO training will result in a slower to teach and less reliable result for some things.