They have some criteria to exclude people from the bad lottery (disabled people and their care takers, unaccompanied minors), and then may apply a ranking based on booking class, frequent flyer status, and check-in time.

But here, the claim is that pax were chosen randomly.

Many people have good reasons to be on a plane, and this passenger being a doctor should not have any bearing on the resolution of the problem at hand (except insofar as it should've compelled other pax to take the 800 USD and hotel room and volunteer for disembarkation).