One of the most consistent criticisms of The Lord of the Rings is the question of why Tolkien didn't just have the Eagles deliver Frodo to Mt. Doom and be done with it. The obvious answer is that there would be no story worth telling if he had done that, but that isn't really a very satisfying answer within the story. The Tolkien scholar David Bratman recently responded on his blog to an inquiry about this question in this way:
I mostly like David's answer, particularly the point about moral perseverance. And his third item about stealth has a certain degree of truth to it. But it still isn't a very satisfying answer within the story. But there is one point that he made that I flatly disagree with, which actually presented to me what I find to be a much more satisfying answer.Your question is one that's often discussed on Tolkien bull-session bulletin boards (which I don't read), but it's not been dealt with at any length by Tolkien scholars, because it's not really a very important question.
The real answer, that is to the question "Why didn't Tolkien write it that way?" you already have - because there would be no story. That's no cop-out but a simple fact. This is fiction, remember, and the reader has to accept the set-up. There's more to it than that, though. Intentionally or not, LOTR is a story of moral perseverance against the odds. Constantly in the story, Frodo and the other heroes succeed because they have put forth their supreme effort. If the job were too easy, they wouldn't succeed. For instance, had Frodo not been brought to extremity in the wilderness and come, through that and the long burden of carrying the Ring, to understand Gollum's suffering, he would not have decided to spare Gollum. Merry and Pippin could never have put the Shire ruffians to flight had they not been tempered in Fangorn, Rohan, and Gondor. This may sound like another cop-out, but it's actually a key to the story. Gandalf indicates in a couple of places that the quest serves a purpose in the hobbits' own moral development: when he tells Frodo "Bilbo was meant to find the Ring, and not by its maker," and when he assures the hobbits near the end that they can settle the Shire's affairs: "That is what you have been trained for." The easy solution to the Ring, one might think, is to have an Eagle fly it to Mount Doom, but we already have solutions such as sending it away or actually using it that are easy, simple - and wrong.
But you want an internal, continuity-based answer. There is none, actually. Tempting as it is to consider Middle-earth a real place, there are many holes in its history that the author never bothered, or never figured out how, to fill. (Could an orc repent, and what would happen if it did? is the biggest; Tolkien spent quite some time in later years scratching his head over that one.) I can make a couple of comments on this question, though.
1) Eagles really aren't a taxi service. They're proud, independent birds, and while they may grant favors, you can't just call on them to solve all your problems.
2) Eagles are also wild, dangerous, and serve no-one but themselves. I wouldn't let one anywhere near the One Ring once it's been rendered "radioactive" so to speak by Sauron's active searching.
3) The Fellowship's only hope for success is to come in to Mordor underneath Sauron's radar, so to speak. Obscurity and stealth are their bywords. A Giant Eagle of the Misty Mountains flying directly towards Mount Doom is going to be noticed. The rescue from Mount Doom is possible only because by that time Sauron and the Nazgûl are otherwise occupied.
Bratman says "Eagles are also wild, dangerous, and serve no-one but themselves." I think that is true of the Eagles of The Hobbit, but it isn't true of the Eagles of LOTR. In The Silmarillion, it is made explicitly clear that the Eagles are servants of Manwë. And, though Chrisopher Tolkien mistakenly obscures this fact, the Eages of LOTR (particularly Gwaihir and Lhandroval) are the same Eagles as the Eagles of the Silmarillion. In the chapter on Beren and Lúthien, the description of Thorondor and his vassals soaring high above Morgoth’s realm replaces a statement that Thorondor was the leader of the eagles, and specifying that his “mightiest vassals” were Lhandroval and Gwaihir. Christopher explains that this change was made to avoid confusion with the passage in The Return of the King describing Gwaihir and Landroval as the “mightiest of the descendants of old Thorondor, who built his eyries in the inaccessible peaks of the Encircling Mountains when Middle-earth was young.” (LOTR, 948.) He points out that at the time he did not understand that Gwaihir’s name in this passage was actually changed from “Gwaewar” in 1951 in order to bring it into accord with The Lord of the Rings and that this change should not have been made. (See The Lost Road, 301.)
So the Eagles in LOTR are still servants of Manwë (which explains in part why Gwaihir is willing to follow Galadriel's instruction to go and find Gandalf and bring him to Lothlórien). But it is well known that the Valar were unwilling to interfere too closely with the affairs of Middle-earth, after the disaster that their bringing the Elves to Valinor caused, and the removal of the Undying Lands from the Circles of the Earth at the Fall of Númenor. Even when they sent the Istari, they severely limited what powers those Maia could use. So I think the "true" answer to the old question of why the Eagles couldn't just carry Frodo to the Cracks of Doom is that doing so would have violated the Valar's prohibition against interfering too closely with the affairs of Middle-earth.