That story was all over the headlines not because it was normal. It was in headlines, and in the front of your mind, because it was the exception that proves the rule. That’s not typical.
Regardless, the FDA approval was mostly about assessing safety of the drug in controlled trials and follow-ups. Likewise, the vaccine studies were rigorous examinations of safety and an in-depth search for warning signs that might indicate problems in small populations. The correct interpretation of this approval is to see that it has been thoroughly examined for safety in the field.
Finally, we already have mountains of evidence that the vaccine is effective. That much is not in question after administering an incredible number of doses globally and observing the outcomes. Proving efficacy was the least of the FDA’s concerns at this point.