It looks to me like the CDC assay PCR is about 98% specific. As long as patients who are being retested during recovery are not included in the "new case" designation, which Tennessee does not, then the percentage of new cases resulting from false positives will be a function of general infection rates.
If you perform 10,000 tests, you would expect about 200 cases from false positives. So, if infections are bottoming out and percent positive is just 4%, then yes, about 50% of those could be false positives.
However, in states like Texas where percent positives are now climbing toward 20%, you would expect false positives to be only about 10% of daily new cases. That's not a small amount, but is much more acceptable than 50%.
I do not know if we are still using the CDC PCR assay, do you?
There are other PCR tests that have shown to be 100% specific.