PCR has a very low false positive rate. Its supposed “high false positive rate” is actually sensitivity for several weeks after previous infection, even after full recovery.
This isn’t a problem when infections are rising because during growth the new, active infections that actually matter always greatly outnumber the old, defeated infections. This enables PCR testing to anticipate hospital demand by about one week.
PCR’s “high false positive rate”, or rather its belated detections, become a problem when infections are falling by wasting the resources of any trace-and-test programme, swamping the signal under noise, and prolonging widespread lockdown longer than needed.