“Note that in a population where few people (e.g., 5%) actually have antibodies, positive results on a test that is not at 100% specific (such as the test from Healgen that provides 97.56% specificity and 100% sensitivity) may only be correct 67.8% the time, although a negative result would nearly always be correct.”
There are a lot of lower specificity tests than the one mentioned above.
False positives are the problem. There are comparatively few false negatives. I believe you have this paradigm confused.