There are many variations of this myth: positive reinforcement doesn't work for "tough" breeds, positive reinforcement doesn't work for "red zone"/aggressive dogs, and so on and so forth. Here's the thing. Positive reinforcement always work if done PROPERLY, and if you are using something the dog actually finds REINFORCING. The behavior sciences are nuanced and complicated; Bob Bailey, a famous and very well respected animal trainer, has said "Training is simple, but it's not always easy". If I'm working with a family that says "positive reinforcement didn't work for my dog" usually it comes down to a few things:
1) They are working the dog over threshold. If I am working with a dog who has a negative emotional response such as fear or aggression my first order of business is to keep the dog from going OVER THRESHOLD, which is when the dog starts to behave aggressively or fearfully. Trying to throw treats or toys at a dog that is already barking and lunging isn't going to change their behavior, but that's not positive reinforcement's fault, that's the fault of poor technique!
2) The owners expected a bit too much too soon. Want a dog with a reliable recall even around prey animals and other dogs? That's awesome, but be prepared to work for quite some time and manage your dog closely using leashes and long lines until the behavior is reliable. Look at it this way: you would never expect a child to learn their alphabet, and then two weeks later be writing a PhD dissertation, first they have to learn about spelling words, then how to write a complete sentence, then paragraphs, and after years and years and YEARS of work they can write their PhD dissertation. But with our dogs we expect them to be able to come when called at the forest preserve off of squirrels after we've called them to come from 10 feet away in the kitchen for a week or two. That's a HUGE skill level gap. Remember, the more difficult and more complicated the parameters for a behavior, the longer it will take to BUILD that behavior!
3) They weren't using something that motivated the dog. Most dogs are food motivated. In fact, all dogs are food motivated to some degree because... they eat! However if there is something in the environment that's a competing motivator such as other dogs to play with, squirrels to chase, people to greet, you have to make sure to have something that the dog finds MORE motivating (this is where I often recommend breaking out the "human food", or if your dog has a favorite toy using that), OR you have to manage the environment so that you can use the competing motivators to your benefit (i.e. teach your dog that they CAN chase the squirrel, but only if they do XYZ behavior FIRST). This often takes some creative thinking, but it is possible!