Gwaihir to the rescue .. again?

[The Saucepan Man]

Here's something that's just occurred to me and I'd welcome any thoughts.

JRR devised a wonderful plot device to get his characters out of trouble - Gwaihir the Windlord, Lord of the Eagles. First he and his kin rescue Bilbo and the dwarves, then Gandalf from the Tower of Orthanc, and finally Frodo and Sam from Mount Doom. So why not use him to help complete the Quest?

When they are in Rivendell, why not summon Gwaihir and get him to carry the Ringbearer to Mordor, fly over Mount Doom and drop the Ring in? Yes, there were nine Nazgul flying about, but I'm sure the Eagles could have distracted them while one flew over Mount Doom. And they could have flown high to avoid Orcish archers (and Sauron's notice?) until they reached Orodruin.

I'm sure there's a perfectly good explanation, since nothing happens (or doesn't happen) in JRR's works without there being a good reason. Can anyone enlighten me?