If we want to ensure that the active population is always above herd immunity threshold, 10% or 20% false positives would be acceptable, since the population in contact would still be 80 or 90% immune.
I think this is more about finding hotspots. Where could a seeding event lead to a large outbreak. Clearly 1 person entering into NYC with COVID today would not have the same effects as 4 months ago because supposedly 20% are immune. At least for the time being at least the assumption is. So there is that effect, but also how big could the next wave be based on that. Of course that all has to be proven that detectable immunity or levels above a certain threshold will prevent you from getting infected again or at least reduce the severity over some period of time. So how many antibodies, which type, how long they last, and how much do they have an effect on reducing reinfection or severity of the next infection - all yet to be determined.
Unless everyone's immunity stops at about the same time, a year or two of immunity would be sufficient to prevent large scale pandemics from recurring from this virus.