A lot of people have burning questions about the Lord of the Rings trilogy. An imgur user, hobbitladyslayer, ventured to answer one of the most common questions: Why didn't Gandalf just fly the ring to Mt. Doom and drop it in?
Inspired by this fan theory on the subject, hobbitladyslayer works very had to destroy my hopes and dreams.
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 neutral beings, thus they are not constrained to do good and are not bound to the same fates as the other races of light (like Elves, Men, etc). Thus they are self-serving, and like the Ents, could only have been compelled to join the battle out of self-preservation or anger. For example, 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).
Also, (referring directly to this theory) Aragorn, Frodo, Gimli AND Legolas would have known who Gwaihir was. So all Gandalf would have had to say was "Oh guys, sorry I'm dying but Gwaihir is totally going to meet up with you, don't screw this uuuuuuuuuuuuup!" as he fell. Saying "Fly, you fools!" to explain all of that is stupid. Plus Gandalf tells them to "Fly" on multiple occasions before this point, but using the definition of "hurry your asses!" not the modifier "use the Eagles!" Sorry. Doesn't quite work.