Why does "no aircraft cross directly over the pole"?
This article from AA (titled "Over the Top" by Gerard J. Arpey) says,


  By the way, because of the limitations of older navigation systems, none of the polar routes we fly crosses exactly over the North Pole. At the Pole, an airplane’s compass changes from a due-north heading to due south, and that change of course could potentially lead to problems with ­earlier-generation autopilot systems. Fortunately, that is not an issue with the latest generation of long-haul aircraft, such as the Boeing 777s we fly, whose source of navigation is the extremely precise Global Positioning System (GPS). Nonetheless, to make polar flying equally effortless for both the new generation and the previous generations of aircraft, none of the approved civil polar routes comes closer than about 60 nautical miles from the Pole.