Hi. I've been researching pet nutrition ever since I owned pets and what I learned is -
Premium dog foods are designed to be nutritionally complete and balanced. That is - everything the dog needs is present in the dog food at a specific balance that is the most optimum for your type of dog and the dog's stage in life (e.g., puppies have a different balance than adult dogs, small breeds have a different balance than large breeds, etc.).
Therefore, offering supplements, unless the dog is exhibiting medical issues - will only upset this balance and would, therefore, not be beneficial. If a dog has medical problems, then offering a supplement to combat that specific issue may be necessary at the advice of a veterinarian. Also, offering other foods as treats (human food, pupperonis, etc.) that is not within the same brand as the dog food at a significant amount can also upset this balance.
This is also true for other pets. For example, I have pet rats as well and I feed them premium Mazuri 6F rat food. Offering other foods like vegetables and such in addition to Mazuri would upset that balance as well.
Now, this is just from my own research and is what I'm following for my own pets. I am no pet expert that's for sure.