I don't know what algorithm they used to select him. But united does have a procedure based on ticket class, time of check-in and other parameters.
If you define it the second way, the doctor hadn't completed boarding. (And I don't think it's a totally ridiculous torturing of the words to read it that way.)
1. Airlines in general don't seem to have defined 'boarded' in their contracts. This usually means that the meaning as tested in court will default to that commonly understood, i.e. 'has entered the aircraft and is seated'.
2. Accepting the manifest isn't final, there have been many instances of aicraft returning to the gate for manifest changes. Even to embark a late but high-value passenger. So when is the aircraft finally boarded? when the wheels leave the ground?
3. The United CEO stated that the flight in question was 'fully boarded'. That may have been an error in terms of hsi legal team but shows that even the airline doesn't have a concrete internal definition.
In Europe even before 9/11 flight security protocols demanded that a passenger need to be present on the plane and be seated until boarding is complete.
No piece of luggage is allowed to go on the flight if the airline / the pilot / crew / airport staff has knowledge that the passenger is not on that flight: They need to open up the cargo area, remove all containers until they find this person's luggage so it can stay on ground.