It's the "a tiny fraction of a percent might possibly have some adverse effect, so therefore I don't believe that any should be used, despite the fact that it can be very easily proven that not having them can have much more serious consequences in very many more cases" attitude of mind, one which twists logic inside out.
I quite agree. The risk of death from automobiles is far greater than the risk of injury from vaccines. Should we ban all cars/ More people die each month from house fires than in a decade from vaccines. Should we ban all houses? More people die each year from peanut butter than from vaccines. Should we ban peanut butter? Should we ban electricity?
Part of the problem is that vaccines have been so incredibly effective in removing the scourge of the diseases. Without the ever present danger people soon forget what they are defending against.