That assume perfectly rational reactions. Many people can't deal with "You tested positive for X. We should keep an eye on it and see if it develops into something." It makes them nervous. They want a pill. They want surgery. etc.
The problem with even really good tests that test exactly what you want is that they have 4 modes-2 good: test positive for X/you actually have X, test false for X/you don't have X and 2 bad: test positive for X/you actually don't have X and test negative for X/you actually do have X.
The problem is that when the actual instance of "you have X" is very low, the "test positive for X/you actually don't have X" can swamp your signal.
Add in the natural noisiness of biological systems, and you wind up with lots of incorrect assessments.