http://www.youtube.com/watch?v=6ZrJPiq9QGM&feature=youtu.be&list=FLNV8O8FCx8YEgx317gHaaQw
It's a bit silly but explains the most obvious reason the Eagles wouldn't work even if they agreed to do so (and Nazgul with Fellbeasts aren't even in the clip).
It's also pretty entertaining 
The eagles COULD fly higher. Still entertaining though.
As the Lord of the Winds, himself, I completely abhor this sentiment. We Eagles would not fly so low nor so slow to be hit by the enemy that easily. I scoff my feathered wings at this video.
That being said...this is still very funny.
But the real reason we didn't fly Gandalf and company to Mt. Doom to destroy the Ring is because we were too busy playing our favorite card game at the time. I remember I had just finished the second stage of The Redhorn Gate when the Fellowship set out from Rivendell. By the time Frodo had reached Mt. Doom we finally were able to collect enough victory points to complete the quest (those locations kept being discarded as shadow cards!). We then flew to the Black Gate just in time to rescue our Heroes. To this day we haven't played that quest again...
Here's better theory why they didn't fly with the eagles but not as entertaining http://www.tickld.com/x/this-guy-just-changed-the-way-we-see-lord-of-the-rings-mind-blown
That explains it all! This **** fools should have just listen what he said! Literally! (Not what they think they heard) I mean watch it and hear for yourself:
http://www.youtube.com/watch?v=6ZrJPiq9QGM&feature=youtu.be&list=FLNV8O8FCx8YEgx317gHaaQw
It's a bit silly but explains the most obvious reason the Eagles wouldn't work even if they agreed to do so (and Nazgul with Fellbeasts aren't even in the clip).
It's also pretty entertaining
This video made my day yesterday. Good find!