For example, the vaccine used for the 2009 Swine Flu:
https://en.wikipedia.org/wiki/Pandemrix
"Pandemrix was found to be associated with narcolepsy from observational studies, increasing the risk of narcolepsy by 5-14 times in children and 2-7 times in adults. The increased risk of narcolepsy due to vaccination in children and adolescents was around 1 incident per 18,400 doses."
If you don't have a large section of the population unvaccinated, you can't conclusively prove that the vaccine was the cause.
It’s just, which would you rather have: a perfectly run trial because your vaccine efficacy is so low that your control group doesn’t feel incentivized to get vaccinated? Or a vaccine that works well enough to quell a pandemic, causing your control group to jump ship and potentially lessen your statistical power?
(disclaimer: i've run experiments with patients, i'm well aware of how messy the world is)
No, you are not. Not even close.