The most important thing is to find participants who are ill. It is better to have a certain percentage of false positives at home as than to have all the people walking around without know they have Covid-19. A large part of the cases is probably still not found, because a lot of people do not show any symptoms.
Ultimately, this is again a statistical problem. Just like with the masks. Nothing is 100%, but to avoid overburdening hospitals with Covid-19 and keep the death number low, it doesn't have to be perfect.