They booted paying passengers for crew members.

They argued that this would result into a cancelled flight the next morning if they wouldn't have flown the crew, since no crew would be available. This may, or may not be the case. But if it really was that urgent they could have chartered a private plance for the crew. At ~300 miles even renting a car may have been feasible (this may have cut into the mandatory rest period of the crew, though).

  Another mistake made was allowing the passenger to board and then removing him

Precisely that! You just can't have paying passengers board a plane to then kick them out. If the flight is really oversold you should find volounteers at the gate and - if necessary - auction the bumping off. If nobody wants to be bumped for 800$ then raise the price. It's not my impression that they really did that. They just selected four passengers, stormed the plane and essentially told them "off!" When Dr. Dao refused (which I can understand, if you have patients potentially waiting for month to get to see the doctor) they gave him the "Nice face, would be a shame if something happens to it" treatment.

Oh, and offering the amount in cash and not in some vouchers, which probably come with so much fine print and restrictions to be virtually useless would probably help too.