I'll tell you why the air defense 'systems' failed; they're not physical systems like a SAM site, they're human and organisational systems.
In a non-war context, noone was ready for such an attack. Paper contigency plans don't help or count. New and unplanned decisions must be made by people. This is a classic management problem, and you'll quickly find that around half of all management decisions taken are actually 'wrong'.
These decisions are based on information. This is inter-organisational stuff. Who knows what's happening? The FBI, the FAA, the ATC services, the 911 emergency services, the military, DoD, the White House? Probably little bits with each, but does anyone have the bigger picture? No. Do these organisations cooperate well? No. Is information sharing, even in an emergency, going to be quick, fluid and cohesive? No.
I'll bet there were individual failings, and mistakes that could have been avoided. I'll also bet that the bigger problem was lack of communications, standard operating procedures, and divides in organisational culture.
If someone takes their plane into unauthorised military airspace or into an airport's departure path, something will be done about it bloody quickly, by that one affected party. 9/11 was a different set of parameters; lots of actors, lots of factors. You can't formulate a fitting response without considering them all, and this is slow. Even now, I wonder if it would be any different. Sure, you can put F-16s up and find the rogue airliner, but who decides to shoot it down? Slow again, and if you're too slow, it's over a population centre and it's too late. More difficult than you think.
That's why it took a long time to coordinate a response. I bet it doesn't now, but I bet it could still be done faster. Such are the rules of humans, groups and organisations.