One issue I see, at least for passenger flights, is the distance that one would need to taxi to the appropriate terminal (especially if landing from an unusual direction). The difficulties could probably be negated by using more general purpose terminals, which seems feasible with better mass transit and faster people-processing. Also, could lead to increased response times for emergency services if something did go wrong.
Also, 2 miles diameter seems rather short to be landing modern airliners on. Even smaller aircraft (e.g. B737, A320) tend to require no less than 1200m (the proper requirements are usually significantly higher) to land safely, and ground steering is not possible until the aircraft is at relatively low speed. Even if the track was banked somewhat (disregarding any aerodynamic/stability issues with this), it seems likely that a much larger radius would be needed. Increased tire wear and lateral stresses on the landing gear are also a possible concern.
Personally, I believe the even-numbered n-gon arrangements that are being suggested by people here retain enough of the purported advantages of this setup that they are what should be considered.