Public health is effectively a multidimensional utilitarianism problem and the mainstream medical establishment chose to de-emphasize legitimate problems, likely because they considered it important to reach high vaccination rates, and any message that would cause people to decline the vaccine was viewed as affecting the rates significantly. Most of the major officials have done their mea culpas, and I think it's high time that we actually invest more into addressing vaccine side effects.
Anecdotally, I was at the doctor's office today and was offered shingrix since I'm over 50. Shingles is terrible, but my doctor basically talked me out of getting the vaccine, by explaining that it had significant and common side effects (sore arm, puffy arm, tired feeling) and that I wasn't really at high risk (other than my age). She also no longer prescribes dual vax (for example, flu + pneumo at the same time), also because of commonly reported side effects (again anecdotally, in my case when I got both, my whole left arm was useless for days and I felt sick, tired, and pained for over a week).