The base rate does most of the work
A 99%-accurate test for a condition affecting one in ten thousand people still yields mostly false positives. The evidence updates the prior; it does not replace it.
What a p-value is not
It is not the probability the hypothesis is false. It is the probability of data this extreme if nothing were going on. Effect size and interval say more than significance.
Regression to the mean
Extreme measurements drift back toward average on their own. This alone manufactures apparent cures, coaching effects, and curses.