Not to mention the perverse incentive: if you feel sick, you want to avoid getting tested and counted as a positive if you don't want your locality or employer to shut down. There is a theory that this accounts for much of Japan's perceived success - antibody studies show just as much prevalence as everywhere else, what happened was that all the mild cases never got tested.
There are also non-perverse incentives. If I feel sick and live alone, a test test result doesn't give me much actionable information, takes time, and puts the people giving the tests at slightly more risk.