United should've planned ahead for their staff to get to where they need without forcing somebody off the plane. They already knew how many tickets were sold, how many checked in and how many boarded (they were already on the plane).
The airline needed to raise the compensation price until someone volunteered, and if still no one volunteered, then tough noogies. They still do not get to shift from a civilized situation to a violent one by dragging state police power into what is strictly a monetary dispute.
In your example the basis for refusing transport is that due to an equipment problem it's not possible for that passenger to comply with FAR safety regulations and therefore is in breach of the contract of carriage for the airline. The terminology of the word boarding isn't even relevant in your example.
Also, boarding is any time up until gate departure.