A lot of people seem to be arguing about what "boarded" means.
But in this case it seems to be irrelevant, as UL was not allowed to deny the good Dr boarding, even at the gate.
CFR 250.2a says
In the event of an oversold flight, every carrier shall ensure that the smallest practicable number of persons holding confirmed reserved space on that flight are denied boarding involuntarily.
This was not an oversold flight, unless we count the DH crew as having confirmed reserved space.
We should then note the definition of "confirmed reserved space"
Confirmed reserved space means space on a specific date and on a specific flight and class of service of a carrier which has been requested by a passenger, including a passenger with a “zero fare ticket,” and which the carrier or its agent has verified, by appropriate notation on the ticket or in any other manner provided therefore by the carrier, as being reserved for the accommodation of the passenger.
Also note the definition of "Zero fare ticket"
Zero fare ticket means a ticket acquired without a substantial monetary payment such as by using frequent flyer miles or vouchers, or a consolidator ticket obtained after a monetary payment that does not show a fare amount on the ticket. A zero fare ticket does not include free or reduced rate air transportation provided to airline employees and guests.
So DH crew, even if booked in on the flight, do not count as a "confirmed reservation" and therefore do not get priority over fare paying passengers. A fare paying passenger can not be denied boarding in favour of a DH crew. To do so would be illegal, no matter what UL's CoC or company policy says.
UL simply can't legally deny a passenger boarding because they want to board a DH crew. Of course they can make an offer sufficiently big enough that someone
voluntarily decides to give up their seat.
In my mind, the definition of boarding is irrelevant in this case, as UL were not legally able to deny boarding to the Dr, in favour of their crew.