It did not assume mutual exclusivity. But if the underlying cause of succumbing to a disease is poor general health or a poor immune system, then preventing a s

It did not assume mutual exclusivity. But if the underlying cause of succumbing to a disease is poor general health or a poor immune system, then preventing a single disease might not really save the person. But it still shows up as a life saved by the vaccine (because it prevented a specific disease incidence). That is misleading. This can be made very clear by taking a small example. Imagine there is an island with a population of 100. Among these 50 are very weak. And every year, these 50 die due to various illness. And among these 50 deaths, 30 are caused by X. So in order to save these 30 deaths, the government started vaccinating against X. The next year there is zero deaths due to X. But there were still 50 deaths in total that year too. Only that the 30 lives that used to be claimed by X, is now claimed by A, B and C (they are so weak that they will succumb to any infection). This is not an unrealistic scenario. If you have a weak immune system, any minor infection could end up fatal. The only unrealistic thing in this story is the occurance of constant 50 people who have weak immune system every year. But that does not void the point. So while the vaccine prevented disease incidence, it did not prevent any deaths.