Perry's breeder uses that approach with her pups......and Perry was the absolute best Basenji I have ever had in terms of being reliable in the house. He didn't even take tissues from the wastebasket! Also, all her dogs learn early what is available for chewing and what is not. Again, Perry never trashed his plush toys. I could leave them with him and he never, ever chewed one up. His breeder tells me most of her pups are the same in their new homes. Anecdotal evidence, to be sure, but certainly my boy's behaviour was exemplary.
IMO, "all positive" is leaving out the important concept of consequences. Aversives need not be severe, but they do need to be unpleasant enough that the dog prefers to avoid them. In nature, dogs learn what is pleasant and what to avoid when they have negative experiences. So do humans, come to that! However, the most important element of training, whatever method you use, is consistency. You can't have one rule today and another tomorrow, but if you draw a line the dog will learn what is acceptable and what is not.