This logic is not necessarily true. The tests are inherently probabilistic. Medical professionals must use the impact of a False positive or negative and the baysean probability of a disease given their test results to determine the best thresholds for patients. False positives can still be very harmful to patients (surgery, medication, lifestyle changes, etc) so they must be taken into account as well.

One classic example is a pregnancy test. You're going to find out eventually, and the impact of a false positive is huge, so the test errs toward false negatives instead of false positives.

This video does a good job explaining it (and is where I learned most of what I know): https://www.youtube.com/watch?v=UF1T7KzRnrs