Yes the "vaccine problem" could be solved by simply removing liability protection. This would force vaccine makers to do proper safety testing and to properly disclose those findings.

This would allow the public to make informed choices. If you had a 1/50000 chance of developing a heart arrhythmia however it protected you against some major pandemic....you could choose the path of the lesser evil.

There are certain tradeoffs that probably wouldn't be worth it....for example when my kids were young I was talking about this concept to his pediatrician (who was not a vaccine pusher)....and the Dr said "basically the risk of severe reaction to chicken pox is fairly minimal and thus might want to just risk it if vaccine side-effects are a concern"