Earlier quoted context omitted.
Or you could flip that around and make testing so plentiful everyone is tested 3x (perhaps by independent labs), so the FP rate is more like... 1/10 * 1/10 * 1/10 (1/1000)
No good reason for test errors to be uncorrelated, ie. quite possible same people who trigger false positives on the first test would be false positives on the second etc.
T1. Drive up saliva swab by nurse, for serology test.
T2. Re-deploy census peeps to get samples (and census info) from entire neighborhoods.
T3. 23andMe-like home kit express mail to lab
The odds of people getting a FP on all three might not be 1 in 1000, but it probably isn't 1 in 10.
Anyway, as it is, leading 1 in 10 people to falsely believe they are immune, isn't much better than having no test at all.