I agree that keeping the immune system well-functioning is very important.

However, from the same link as above:

"Professor Coles has done 11 autopsies on supercentenarians and finds that most die of congestive heart failure secondary to “systemic TTR amyloidosis”, a thickening of the blood. The rest tend to inhale food particles and get pneumonia. It is not really clear why women live longer than men; probably something to do with their having a different cocktail of steroid hormones."

Amyloidosis is basically proteins which are normally soluble becoming insoluble and accumulating in intracellular space, etc. That's essentially wear and tear, not an immune system failure. On the other hand pneumonia is a pretty clear immune system failure.