I dug into this shortly after your posting but didn't quite get to a finished result. Â From the MIT article, follow the links to the referenced reports. Â There you'll see the procedure and results of testing of various PCR testing kits.
Basically, it seems the tests are very specific. Â Apparently different PCR test kits may detect different parts of the virus so it is possible that false positives might differ for different test vendors. Â Although one report suggests that other coronaviruses might give false positives, none of the reports that I saw talk about actual false positives, where the result would be repeatable. Â None showed any false positives.
The testing that was being done had results of Positive, Negative, Inconclusive, and Invalid.
It seemed to me that the more likely cases for a false positive would be from mishandling of the sample, perhaps contaminating it at collection or in processing, or by some labelling or data entry error. Â Those kinds of errors are not something where the statistics seem reliable. Â For instance, if sports team A and sports team B had different people doing collection and different labs, I wouldn't expect a 0.01% false positive rate for one team to apply to the other team.
I think it would have been instructive to include known false and positive samples with each group of samples to get some idea of FP and FN samples. Â I expect the labs run those on their own (or, as part of what is required of them).
There is one researcher who has been studying & publishing on COVID false-positive rates. Â He indicates that FP rates are somewhere roughly in the 0.5% range. Â As he points out, when positive rates get low, the percent of the positives that are false goes up.
November 2020 short article  Unfortunately, he talks of studies that provide his external quality assessments but gives no references.
August 2020 paper Â
May 2020 paper Â
At that time, there were no external quality assessments for COVID PCR tests.
Currently, Santa Clara County is showing 1.7% positive rates. Â If 0.5% of the true negatives have results as false positives, then that is a significant percentage of the positive rates. Â
However, we don't know whether two or three samples are being. Â If the errors are independent on different samples, then taking & processing two samples results in FP rate of 0.5% dropping to 0.0025%. Â If the errors are not independent, then this redundancy doesn't help.