https://www.cdc.gov/coronavirus/2019-ncov/lab/resources/anti...
>The first vaccines distributed in the United States induce antibodies to S protein. Thus, presence of antibodies to N protein indicates previous natural infection regardless of vaccination status, while presence of antibodies to S protein indicates either previous natural infection or vaccination.
The other wrinkle (and possibly the most important one) is that we don’t want to create a system where antivaxxers have an incentive to deliberately infect themselves and/or others.
The FDA approved Roche serology test has a specificity greater than 99.8% and sensitivity of 100% (14 Days post-PCR confirmation) https://www.roche.com/media/releases/med-cor-2020-05-03.htm
I wouldn't say 0.02% is a particularly high FP rate.