Those pesky eagles: explained
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Many wonder why the Great Eagles didn't just carry Gandalf and Frodo to Mt. Doom to destroy the ring, thus saving a lot of time and lives along the way.
In Middle-Earth, Tolkien describes the Eagles much like the origin of the Ents. They are beings without Fëa, or souls, if you will. Without Fëa, they are not constrained to do good and are not bound to the same fates as the other creations of Illuvatar (like Elves, Men, etc.). Thus they are self-serving, and like the Ents, could only have been constrained to join the battle out of self-preservation or anger (after destroying Isengard and imprisoning Saruman the Ents went back to their normal business, they wouldn't have marched on Mordor because they couldn't see the danger).
They were proud creatures, and do not enjoy carrying burdens, seeing themselves above the "pack animals" of the world, only occasionally helping Gandalf because Gwaihir, the Lord of the Eagles was once healed by Gandalf.
To help Gandalf by flying them across Middle-Earth would have likely been seen as unneeded, or below them. And they likely wouldn't have seen the urgency or necessity of the ring's destruction (few people outside of Gondor even acknowledged the danger of Sauron regaining power).
Plus the obvious: the eye would have seen them.
source:
Relatively Unknown LoTR Fact that People Might Wonder About - Imgur