The real proof is history.
The real proof that vaccinating children and adults works is found in history. Until vaccination became standard children and adults died or were disfigured by diseases that are now preventable. Yes a very few may have a reaction, but when you consider the huge numbers of people who have never had to get these diseases, it is worth the risk. At one time it was common for people to have serious scarring from even chicken pox. Measles left people blind, mothers who gave birth to children with birth defects. Mumps can cause sterility in men. Obviously these people have never lived when there were polio epidemics. The fear was not unlike what we experienced as a nation on 9/11. That is as close as I can come to an analogy. Do we really want to go back?