I completely reject your premise that the covid statistics are shoddy. It's easy to confirm a covid case. You do a PCR test, and if you get a positive back it's covid, end of story. It's admittedly a bit trickier to attribute a covid death... but not really. If a covid patient dies of a heart attack, it could be bad luck, or it could be the disease that coagulates its victims' blood. But usually it's pretty obvious. Someone dying of pneumonia and/or a cytokine storm isn't exactly subtle. There's nothing in the overall death statistics that suggest that the obvious covid deaths are anything but the obvious.
By comparison, establishing a casual connection between a vaccine and any adverse side effects is incredibly difficult, simply because they're so rare. Of the 44000 people in the Pfizer trials, literally none of them died of covid OR the vaccine, vs 15 covid deaths in the unvaccinated control group. And VAERS is just a safety net in case something small-but serious slips through the statistical power of a full scientific study (as happened with the J&J vaccine).