Are you serious? It is an assay of neutralizing antibodies and that is quite specific and has excellent predictive properties for the real world. How do you think monoclonal antibodies that are prophylactic were developed?
We have plenty of People infected daily, it's a simple question "Is this the 1st you were aware you have been infected?
They can tell us every 5 minutes about who wasn't jabbed.
Yes there are easily accessible records of who has been vaccinated. You can't just access someone's personal record of who was infected let alone if they were reinfected. What do you think HIPAA is all about?