False positives are bad, but the damage from a false positive is minimal. You’re told that you have the disease even though you don’t, so you quarantine yourself. Two or three weeks later, you reemerge. That’s costly in terms of productivity but you’re not spreading any viruses that way.
A false negative, where you’re told that you don’t have the disease even though you do, is a different matter. The Wall Street Journal cites a specific case in this story of a mom in Texas who came down with a fever and cough, isolated herself in the family home, then tested negative. So she began mixing with her family again. Now her daughter has aches and fatigue. Is that COVID-19 or something more mundane, like the flu?
Every plan circulating right now for reopening the country after this first epidemic wave passes makes widespread reliable testing its key prerequisite. We put out the fire (mostly) now, then we go back to work in the expectation that any new emerging hot spots will be quickly identified by easily accessible tests and contained.