Continuity mistake: As the goons are trying to get the van to stop, it changes from a Ford van to a Dodge with a messed up bumper, and back to the Ford between shots, and the bumpers don't show any damage.
Continuity mistake: The number B.A. gives Hannibal is 555-2944, but the number Face looks up is 555-7477.
Answer: Several times, Face would pose as a doctor to get Murdock out of the Mental Hospital. How easy would it have been to convince airport officials that he was a doctor taking a sedated patient on board? Or simply tell them, this man is afraid of flying, so he had to be sedated. They could say, "Believe me, you don't want this man to be awake."