You are right that there IS a problem with PCR testing: it is that you might show positive weeks or months after having COVID. So if it is being used to quarantine someone, or as a determination as to whether you can get on a plane, it may effectively show a false “contagious” positive. It could show you as sick when that is clearly no longer true because it is amplifying old genetic material. But the one thing it is effective at is showing whether you’ve had COVID at some point, so it is a reasonable way to count total cases. There will be a low false positive for total cases, and high false positive for currently sick.
I dont think that is totally proven because there are reports of PCR tests finding fragments of COVID which doesn't prove it was actually contracted or different PCR results showing inanimate objects with COVID due to high PCR cycles.