I don't think you solve much of anything only testing if someone has it.
You also have to test whether they've had it. There may be fairly large numbers who were asymptomatic or had symptoms that weren't enough to go to hospital.
Assuming that 99.99% of folks don't get it twice, then you can identify the people who can go back to work.
That will be the way to get folks out of isolation and the economy running again. It's going to be a gradual process, if done right.
At some point, you isolate the pockets where there are still a few people with it and let everyone else out, including those who haven't had it. Finally, the disease is flushed out of the pockets.
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