What are you talking about? Let's assume that the tests had a really high false positive rate. That means many more people are forced to isolate than required. If everyone truly did what they were told, and the virus was truly being eliminated, then this just means less and less people are getting sick and presenting for testing. After a while, the death rate related to COVID drops through the floor, and the only remaining false positives are people who contract a flu/cold that with similar symptoms. The false positives in those camps will have a significantly lower death/ICU rate to COVID.
The testing reliability does not affect the growth rate, and that is the final arbiter on whether a spread is containable or not. Contact tracing follows the infection path of a particular person, so these 'false positives' have to persist along the chain or they just appear as noise in the general population. At that point, with no perceived growth, and non correlated ICU/deaths, the tests can be reapplied to verify new cases more accurately.
You just have to look at the results in Australia. Tens of thousands of daily tests, and only a few positives - and recently zero positives over a few days. So the false positive rate is incredibly low if anything. And each case can be watched like a hawk now that the numbers are manageable.
So your assertion that any false positive rate means a mathematic impossibility of an end to the pandemic is not supported. Also, the false positive rate is incredibly low, and definitely low enough that the virus can be detectably contained to a point where it is theoretically possible for the pandemic to end.
That doesn't mean it will, simply because 'hidden clusters' may keep it alive until a vaccine is available. But it is not because of the false positive rate of the tests.