I will take a crack at this, as no means a LOTR expert, but a very dedicated enthusiast (I reread it every year).
Well yes, this is a "plot hole". But the point of the story (even the bastardized, though good in its own way, movie version) is not about the destruction of the ring, not really. The point is much more about the journey; at its heart it is really a "coming of age story" of a kind. It is about the seemingly weak versus the frighteningly powerful. It is a story of good versus evil, and the shades of gray in between.
That is a brief explanation in terms of the story itself.
Lore wise the eagles are not an express service you can call upon like a taxi. The eagles are (in)direct servants of Manwe (chief of the Valar, i.e. gods) and appear when hope and victory "needs" to be turned toward the 'good guys'. In The Hobbit and LOTR there is the parallel eagles appearing at the climax of a battle to help turn the tide in the favor of the protagonists. Similarly Gandalf is saved by the lord of eagles (after he dies from his wounds fighting the balrog on top of the mountain) because Galadriel is looking for him, and Manwe has just "sent him back" to the world so he can continue and fulfill his duty in Middle Earth. Before that Gandalf is saved from the top of Orthanc by the lord of eagles because he has not yet fulfilled his duty. These instances are not random occurrences that mean a troupe of eagles can just fly the Fellowship into Mordor and drop the ring into Mt. Doom.
I hope that helps answer your question, and you can go back to enjoying the books and movies.