The breakdown appears to have been with the four crew showing up at the gate, unannounced after boarding had started. They should have been told, and UA's internal policies established as, they were "too late" for a flight where denied boarding must be implemented as a result.
Passengers must show up at the gate on time. Yes, I know that "stuff happens" and it probably was outside the crew's control, but having a deadline set out in the manual would have avoided the situation.
West Coast, airlines are not limited in what they can offer a volunteer. The compensation for a denied boarding victim is capped, but only company policy can limit an offer to a volunteer.