"You don't hear about doctors being given a mystery patient and forty-five minutes to save their life using only a whiteboard"
That's actually a pretty accurate description of what doctors have to do, at least in the UK. For the first nine years after finishing medical school, doctors have to take a series of exams, including written exams and practical tests, which are primarily diagnosing and proposing treatments for real ("mystery") patients. This is a centralised process, rather than ad-hoc tests at every interview, but doctors do have to continue demonstrating their competence during their careers.
The centralization is the critical difference, though. Lots of professions have ongoing exams. Programming is different in that there's no institution to set the exams, so instead we have ad hoc tests of dubious reliability, repeated for every interview.
That's actually a pretty accurate description of what doctors have to do, at least in the UK. For the first nine years after finishing medical school, doctors have to take a series of exams, including written exams and practical tests, which are primarily diagnosing and proposing treatments for real ("mystery") patients. This is a centralised process, rather than ad-hoc tests at every interview, but doctors do have to continue demonstrating their competence during their careers.