First, we should not let public policy be guided by the most risk averse, fearful, and scientifically illiterate among us. Letting the fearful lead is a recipe for stagnation, mediocrity, and eventual collapse. (N.b. these categories are neither mutually exclusive nor perfectly correlated).
Second, there is no guarantee that the risk averse, fearful and scientifically illiterate will be convinced by extra FDA investigation and there is plenty of evidence that they won’t be. Dozens of well-done studies have found no link between vaccines and autism. The scientific evidence that vaccines don’t cause autism is very strong. Yet many people don’t care. Moreover, I bet there is a significant overlap between those who think or fear that vaccines cause autism and those who fear a COVID vaccine. Will a few weeks of extra FDA investigation win these people over? No. More science won’t end science denialism.
Third, rather than alleviating fear, FDA delay may increase fear. People may reason, if the FDA is taking this long to review the evidence when thousands of people are dying every day it must be a hard decision. Delay also makes the vaccine less useful and less obviously useful. Thus, if vaccines come too late people will say that we were reaching herd immunity anyway and that vaccines are useless.