But, you read about rabies (no cure, horrible death), and even if it is a 1-10 million chance and you can do something about it — well, he got the shot (over my protest!).
I think this is similar — child abducted and god knows what happens to them? And it’s your fault as a parent for not supervising? Even a 1-in-10 million chance seems like too much.
It’s not rational, but I think that’s the psychology. It is countered by mentioning the side effects of the vaccine —in this case, identifying the potential harms of over-supervision.