You know the story of the boy who cried wolf, right?
It's the same idea. After some classifier gives false alarms (cries wolf) so many times, the medical staff will turn it off or ignore it.
"Oh, this again! NOPE!"
At least with the bioengineering group I've worked with, the emphasis is on reducing FPR specifically because the goal is to make a tool that will alert physicians to potential pathology, and they've told us that they will ignore a product that cries wolf too much.
For a product that aids physicians, we have to appeal to their psychology, despite the legitimate argument that missing the wolf on the farm is worse than crying wolf.