Here's an article that explains how a false-positive rate of 0.5% and an infection rate of 1% can result in 40% of positive results being wrong:
https://www.icd10monitor.com/false-positives-in-pcr-tests-fo...
They do get false positives, about one in ten or twenty thousand it seems, but most of them had the virus at some point in the past and the test was still detecting dead virus fragments, so the "completely false" positive rate is even lower.
The false positives are re- tested, and it's almost always concluded that they're false. Hence 61 days with no cases - any false positives in that time have been identified as such, so the process taken as a whole has a very low false positive rate indeed.
But even the raw "one in 10k" false positive rate they see is low enough that almost all positive tests anywhere with an active COVID outbreak are true positives.
If our rate of tests produced the rate of false positives you claim are possible, what is the reason for the excess deaths across the globe? That occurred simultaneously, and in a delayed rate right along with positive test rates?