I think that is true, but we can still distinguish between the false positive rate of the testing mechanism vs. the false positive rate of a particular series o

I think that is true, but we can still distinguish between the false positive rate of the testing mechanism vs. the false positive rate of a particular series of tests (where this would depend, somewhat, in the underlying distribution). I'm not sure what the correct terminology is for this distinction, but I still don't believe Bayes' Theorem would come into play.