There is that but I think a lot of people who are anti-vax have reached that stage through personal negative experience. That's why I'm sympathetic. If my kid had a bad reaction, I'd also freak out. But a lot of people don't try to understand both sides and weigh the matter rationally.
Negative experiences do not equate to scientific evidence. Now yes, some children can potentially have a "negative" reaction to vaccination or appendectomies or anesthetic or even water. That's just a reality. Nothing, not even in Medical Science can ever be 100% foolproof. Due to differing chemical reactions one's body might have, you will always have some side effects of anything to do with medication. Fact of life.
I am sympathetic to those parents who experienced a negative reaction on the part of their kid/s and vaccines. That is unfortunate. I wish that upon no parent, ever. The thing is though Vaccination by it's very nature is one of those "for the greater good" things.
The only way to weigh on this particular matter rationally, is unfortunately to disregard (at least intellectually) those bad reactions. Because at the of the day the equation actually is one kid dying/having a bad reaction vs potentially thousands upon thousands of children dying or suffering horrifically from a completely preventable disease. Which one is more desirable?