Honestly, I never found a huge difference with my cats or dogs in this respect and I did try the organic/natural dog foods. BUT, those can actually be worse because some of them have more wheat and that is not very healthy for dogs, cats (or humans) and I honestly think some of those foods were the cause of one of my cat's liver disease.
I have friends that have had really good luck giving their dog supplements for the joint pain/inflammation. KVPet has quite a few. My breeder also recommends the Petsaver vitamin product - one dog that I got from her is outrageously healthy for the breed - she had 7 healthy puppies and delivered them naturally when typically corgis need c-sections. And, she actually recommends the chicken dog food that you can buy at Sam's. I've also heard of several vets that actually recommend the Walmart brand. If you read the labels these products are equivalent to the specialty brands. That all said, there are exceptions. I had a cat that had severe liver disease and the Science Diet KD formula helped her tremendously. However, one of the reasons I don't think these special foods work well in all cases is that her brother was on the same food for the past 6 years and he just died from kidney failure a few months ago. So, I really think that most of the pet foods are designed to met the nutritional demands of our pets and since they don't typically (or shouldn't) get junk food (which is why many Americans develop health problems), they really are on a fairly optimal diet.
I agree with the responses on the table food and as far as making your own food, you do need to be careful that you have the right balance of protein and not too much grain.
In the end, I think our pets, like humans have a certain life expectancy and the best thing we can do is to keep the comfortable in their old age - i.e., I don't think food is the big contributor at this point. Indeed, it's very well known that it's the quality of food that pets and people get during their growth years that makes a huge difference in long term healthiness and lifespan and health obviously are largely influenced by genetics and biology - i.e., big dogs have shorter lifespans than small dogs, etc.
Good luck and you are very kind to be so thoughtful about this issue.