The OP was a double-header: a puzzle and a consciousness raiser.
Kudos.
The puzzle part teaches that you need a test that has at least an order of magnitude better accuracy than the needle in a haystack that you're looking for. If, that is, you are going to rely on a single test. The puzzle made them equal [1% offenders and 1% error rate] so the results ended up a coin toss. Thus it was noted, if you're working under those statistical conditions, your strategy must include retesting. [Not to quibble over terminology.]
Testing strategies undoubtedly vary among the organizations that employ them. Some do a worse job than others. It's a reach, however, to throw out screenings, solely on that basis. Also noted, if you can't reduce the false positive rate, then enriching the population is indicated -something that retesting accomplishes.
On these points, Major League Baseball's Drug Policy and Prevention Program is an interesting case to examine.
Recognizing that better accuracy may be bought at higher cost, MLB employs a two-tiered test analysis strategy: a more economical and speedier one for screening, followed by a more accurate and costly definitive test on the positive screen results. Here's an excerpt from
Section 6. Laboratories:
If the screening test gives a presumptive positive result, the drug's presence must be confirmed by a second definitive test using the gas chromatography/mass spectrometry (GC/MS) technique.
Regarding the consequences of an initial positive test, MLB does not "rush to judgment," either, as seen from this excerpt from
Section 7. Discipline:
An initial positive test result, the admission of drug use or the identification of drug use through other means will not immediately result in discipline for the player of Baseball personnel involved other than being required to participate in Baseball's testing program. Again, enriching the test population.
This strategy could serve as a model for other organizations.
Bottom line: know what you're doing, but then go ahead.
Kudos to Topher for a great post.