The problem is that side effects are massively over reported. I don't have the original source, but I came across a report on a double blind test of one of the COVID vaccines where 50% of the participants received the vaccine verses 50% who received a placebo (saline). I that particular study more than two thirds of the participants reported side effects. Unfortunately most of the time that people do report side effects they aren't real; which creates bias in society and the medical community.

Administering a vaccine to a large population is rather like the trolly problem, as to protect a large number of people you will have to risk a small number of casualties and deaths.