A shotgun is a good choice for home defense but wouldn't be all that useful in the tight confines of an airliner cockpit. It's pretty cramped up front so pulling out a shotgun and bringing it to bear on the intruder would take too much time. Also, a shotgun is just too powerful for a pressurized fuselage. Even using small pellets could cause a lot of damage to the plane.

A strong door helps slow down whomever is trying to enter the cockpit. That gives the crew time to get the ax. IMO, the only mistake the crew made was in leaving him alive. It's hard to interrogate a dead man but carrying a would-be cockpit intruder out in a body bag would send a definite message to the whack-jobs out there - "Try to force your way into the cockpit and you'll die, painfully and without exception."