Earlier quoted context omitted.
Full FDA approval has happened though. It was basically just a formality that anti-vaxxers managed to find. At this point the vaccines are known to be safe, they have billions of doses given.
There are no way to know the long terms effects (if you do please provide the plans of your time traveling machine to nearest patent office). The drug companies are not only under total immunity but also will get paid regardless if the vaccine works or not in the long term. The reporting of adverse effects is probably getting suppressed, for example OSHA is now admitting it won't record vaccines adverse effects so it…
Vaccines almost never have long term side effects, and there is no reason to believe the vaccines at hand have any reason to have any. But maybe it does. What's the probability? Based on those priors, extremely low. Not zero, but extremely low.
The disease, on the other hand, definitely has long term side effects. We know that.
If the risk of infection was low, you'd have a point. Maybe. But the fact is, without vaccine, everyone stands a very significant chance of getting infected. The very small risk of long term side effects is just ridiculous compared to those of the disease, and unlike the vaccine, the disease does not care about your "freedom."