"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.