And, in addition to possible anti-eagle flak and nazgul air combat, what about
kamikaze nazgul air attacks
, with the ring then easily retreived on the ground.
Plus, recall that even Gwaihir told Gandalf, when rescuing him from Orthanc,
that he couldn't carry him forever, so a Great Eagle would have had to travel in several stages, with perhaps the nearest final flight originating from the White
Mountains?