Running the tests again only works if the false positive rate is just due to noise. I.e. if the occurrence of a false positive is uncorrelated between re-tests. I guess most real false positives aren't due to noise, but due to the test triggering on something unrelated and rare. In that case re-testing will yield the same result.

I would urge you to do some more research. Over testing is well known to cause higher rates of unnecessary medical procedures. There are many ways it can break down. For instance, the test may false positive consistently if an individual has some other factor such as mediation, virus, or bacteria that confound the test. Tests are also only validated under certain criteria. For instance if the clinical trial only tested people who presented with certain symptoms, the efficacy of the test may be completely different on the general public.

Well then all that says is you need better standards for tests.

Just because some tests might have alarmingly high false positive rates doesn’t mean all, or even most do.