Ready to be corrected but I think that a 1st Class pax travelling on a ticket issued 100% in return for airmiles may have a lower boarding priority than a pax who has paid money up front, however, when boarding commences the flight is usually officially 'closed', that can mean no more tickets accepted and turning up with a full 1st class ticket, fare paid after the flight has closed may only get you an assurance you will be firm booked on the next available flight. Staff may try to get a voluntary downgrade or offload, that will be an internal affair.
Once again, for those that still seem to be in doubt, the seating and boarding are entirely the responsibility of the ground staff, right up to the General Manager - Traffic, of the airline. The Captain, F/O and Cabin Crew are responsible for the safe operation of the flight.
It is interesting that the Doctor's lawyers are concentrating on the personal injury and humiliation aspect, a stone cold certainty and where the big money is, rather than involving themselves in boarding and seating disputes which, by comparison, pale into insignificance.