The other reason for having 100% sensitive tests at the cost of specificity is because of the clinical tradeoffs that occur because you have done that test.
If, for example, the treatment subsequent to a test is vitamin supplementation which should have next to zero complications then 100% sensitivity is the face of nasty complications caused by non-treatment makes quite a lot of sense.
In some of the areas that I work, like pain, we are denied these elegant trade-offs. However, I also do work in coagulation and there are good examples of this type of trade-off there. For example, despite the limited evidence of clinical utility, it can make sense for people with Homocysteine and MTHFR mutations to take b-vitamins. Similarly, a few false positives have very limited impact on the patients involved as the risk of taking a b-vitamin supplement (in the first world where economic hardship is unlikely) is small.
So this is a good reminder that there are no sweeping generalizations in epidemiology.