Would this gross example of irrationality be tolerated in other professions?
What gross example of irrationality? The vast majority of people with headaches don't have anything to worry about.
The question is whether "people with headaches" is the right reference class. If the headache is unusually severe or persistent, it makes sense to look deeper. Also, a doctor can ask for details about the headache before prescribing the expensive tests.
If it's worth saying, but not worth its own post, even in Discussion, it goes here.