I understand the attraction, but models are often wrong about stuff.
In addition to saliva tests to determine if someone is actively positive for this virus (infectious), we need to start producing lots of blood tests, to determine if they already have immunity from previous exposure.
By sampling the population to see how many already have antibodies, we will know much better how far along the infection has really spread, and what the mortality rate really is. Otherwise, we are dealing with a dramatically skewed sample, of only the worst cases. As of now, those with little to no symptoms are generally not tested, and not included in the statistics at all.
Also, being able to certify which individuals already have immunity, clears them to get back to work, help others without protective equipment, or travel.