It's extremely rare for bridges to fail. I certainly wouldn't expect a bridge to fail given modern safety standards.
You might have marginally more confidence in the bridge after a few hundred thousand cars, but we have processes in place to be confident within reasonable doubt that things are safe before that.
No one is claiming that anything is "100% safe". That's an unachievable level of confidence. If everyone waited for a hundred thousand other people to do anything we'd never get anywhere.
It probably says something that neither of the ‘new medicines’ you cite (neither of which are or were vaccines) are >30 years old. Thalidomide was banned before most people on here were born.
There’s reason for scepticism in all things - but scepticism for its own sake, and without any evidence other than association is probably less than helpful.
This applies to everything from building bridges (University of Miami), through software products (Mars Climate Orbiter) to designing new medicines (thalidomide / paroxetine).
Arrogance that something is 100% safe is only ignorance. The point is we'd like to understand what the known risks are and the mechanisms are. Some of us would like to see a few hundred-thousand cars go over that new wobbly cantilever bridge design.
This is different from complete ignorance and denial which is a separate issue.