Some trainers rely on the use of treats solely and rarely do any kind of training without their trusty treat bag, while others downright refuse to use treats at all. So its not surprising that there is some confusion on when is the appropriate time to use treats when working with our dogs. Personally, I believe that treats are a very powerful tool to train dogs, but that many situations are better addressed without the use of food.
To shape a new behavior (like teaching a dog to sit, or down), theres nothing better than a tasty tid-bit to lure your dog. Treats should always be the first thing you go to when teaching any obedience command or trick. Its simple, its easy and it works like a charm in almost every case. For those rare dogs that are not all that food motivated, the treats lose their power and we then to try another reward (like toys or affection). However, ninety five percent of dogs will perform quite well when motivated by their favorite snack.
I do not use treats for many behavioral issues though. For example, something like dog reactivity is best addressed without treats. Ive seen people literally throwing treats at their dogs in an attempt to distract them from an approaching dog. Even if the distraction works and the dog ignores the other dog, at best you are putting a very temporary band aid on the problem. Instead, you need to work with the dog at a distance, change his state of mind and him move him forward. Ive found simply walking dogs together to be the best way to work on this with your treats holstered.
The big exception here is for cases of anxiety or fear. Then in addition to be very respectful of each dogs threshold of tolerance to a given stimulus, treats (especially high impact treats) can be used to encourage the dog to work slowly past his fears and reshape his perception to something positive.
Every situation and dog is different, so its not as easy as black and white. As a general rule I always use treats to shape new behaviors and rarely to alleviate a problem behavior. Since I think that raising dogs is very similar to raising kids Ill give you an analogy with my kids. If my one daughter hits her sister and then stops hitting her, Im not going to give her a reward for stopping. The rule is you dont hit your sister and its time for me to do some parenting. But if my daughter refrains from hitting her sister in a situation that she usually does, I would reward her like crazy.
Now lets take a similar situation with your dog. Lets say my dog is chewing the rug and then stops as I come into the room. Im not going to reward the cessation of her chewing, but instead Im going to do some leadership (aka parenting). However, if my dog is a rug chewer and is headed toward the rug and she ignores it on her own or with a mild verbal correction from me, Ill shower her with treats.
Thats my take on the treat paradox. Remember treats should always be used as rewards and not bribes. Treats are great but be cautious about spitting them out like a popcorn machine.