This is incorrect. Vaccines take forever to develop for economic reasons. Big studies are very expensive, so companies are typically quite hesitant to move to the next step. Vaccine side effects are usually quite rare, so you wouldn't see them in a Phase-3 trial, even if you waited for decades. This is why any drug is studied after approval in Phase-4 trials. Furthermore, vaccines side effects have historically pretty much always shown up at most weeks after the vaccination.
