The hard part in the U.S. is that the CDC and FDA don't act as central health-data clearing houses the way, say, the British NHS does. In the U.S., health data collection is fragmented, at minimum, among the 50+ states and territories. Not everyplace reports the same information the same way on the same time frames.
So one state may have a robust, well-maintained vaccine surveillance data collection system. And another state might not.
All that said,
this CDC white paper might address (if not completely answer) some of your concerns. "
How are the VAERS data used?" on page 1 might be of interest.
Reading that paper through also got me wondering whether or not people entering VAERS reports have to indicate their relationship to the patient (broadly: Parent/Caregiver, Healthcare Provider, or Self). Perhaps it's possible in VAERS data to simply strip out or else variably weigh self-reporting. Or isolate on provider reports. Something like that.