"The problem is important in any area where a less-than-perfect screen is used to detect a rare event in a population"
Unfortunately, there is no such thing as a perfect screening test for anything in medicine. Some are better than others, but none are perfect. This is a very difficult concept for most people, unfortunately, and for many insurance companies.
It is not such an issue for the better screening tests such as colonoscopy but it is very difficult for things like PSA where there is a large body of evidence it can do more harm than good on average if used routinely even within the recommended ages. For a patient, you're lucky if you can have a meaningful discussion in 5-10 minutes which is an awful large chunk of an office visit that usually has >4 talking points.
It is a problem for doctors and insurance companies because some well intended person with the insurance company will decide to measure the quality of its doctors (which I support in theory) by measuring, for instance, the percentage of age and gender appropriate patients under the care of a given physician that have their PSA checked annually. The problem is, there is absolutely no concensus in medicine that it should be checked regularly as a screening test. I'm not sure I want mine tested when the time comes around unless my family history changes between now and then. So to measure a physician by this marker or other screening tests is fraught with problems, since many patients might opt out for very good reasons. Also, I'm not going to recommend any test because an insurance company wants me to, only if it is right for any given patient.
Bottom line is there are no perfect tests and testing is not always the right thing to do. Most people do not understand that because it is a hard concept to grasp.