lord of the rings - Why didn't Gandalf or Frodo Fly to Mount Doom?

I'm answering because I think there is an aspect of the use of magic in the LoTR that I commonly see ignored and is ignored in many of the answers.

One does not simply fly directly into the gaze of the Lidless Eye.

Enchantment vs. Magic

I am not a Tolkien expert, but my understanding of Tolkien's use of magic, I believe described in his "On Fairy Stories", distinguishes between a more mechanical kind of magic and a more spiritual "enchantment". Forgive/Correct me if I'm misusing his use of the words. It's been a long while since I've read it. There are many many examples of this enchantment, but some that spring to my mind are Faramir's description of the character of staves that he gives to Frodo and Sam, and of their experiences with the elven cloaks, roap, boats, etc. No concrete rules of physics are established. Instead, objects seem to vaguely bend the rules of physics in accordance with the atmosphere/nature of the object or their creator. And it's often not completely explained or even explicitly claimed. It is noteworthy that we're never conclusively told whether Sam's rope untied itself at his call or whether his knot was bad when they climb down the cliff at night. When the party is climbing Caradhras, they discuss whether the "fell voices" they hear are the wind or some creature. Tolkien likes to maintain uncertainty, and it's more about the mood & (spiritual?) atmosphere than the physics of what is occurring. Another example is the way the book describes the wills of Gandalf and the Balrog when they first confront one another on each side of a door, although that is a little more explicit. Also, when the 3 travelers Gimli, Aragorn, and Legolas are chasing the orcs who have captured Merry and Pippin, they talk of a hostile will that is set against them as they journey that is making their trek more wearying. Tolkien doesn't usually enumerate the way the different people and objects can manipulate the laws of physics like some other fantasy works of literature (in contrast to the use of the One Power in the Wheel of Time, for example). Instead, Tolkien creates "atmospheres" and "auras", for lack of a better word, around people and objects to give them a sort of personality, a nature, a character. It's part of what makes the conflict between good and evil so epic in the story, I think. Tolkien's use of wizardry and sorcery in LoTR is, in my opinion, more along the lines of "enchantment" (as referenced in "On Fairy Stories"), but I consistently see people treating it as a mechanical manipulation of physics. Gandalf is an interesting example, as we can name some specific mechanical feats he can accomplish with fire. But other things he does are more vague and more about who he is than any ability we could name. His power as compared in the story in the eyes of Pippin with Denethor, for example. (This is also why doing things like pitting Gandalf against Darth Vader in a fight don't work well, in my opinion. You have to decide which kind of magic/powers the fight will be composed of, and in my opinion, that determines who will win.) Aragorn is the same way, as is Faramir, Galadriel, and most of the characters in the book.

Sauron's Power is Firstly Enchantment, not Magic

One of the objections I saw raised to the eagle plan was that Sauron could conceivably throw fireballs at Frodo on an eagle, but I think this misses the point of Sauron's power. There is one part of the story where Gandalf says somewhat ominously that he has not yet been tested against Sauron. And Aragorn when he confronts Sauron with the Palantir barely has the strength to overcome him and wrest the stone away. It is mentioned that if the hobbits were captured, there is no doubt that in the end they would tell everything. And I don't think it would be in line with Tolkien to attribute this to merely physical torture. See Pippin's encounter with Sauron through the stone. I think you could attribute parts of this to some mechanical ability, but not all of it.

This is one of the beautiful things about Tolkien's writing that the movies could not or did not fully convey, I don't think.

The Eagle Plan Doesn't Solve the Real Problem of Sauron's Strength

I think the main reason why the eagle plan would not work is that few if any on Middle Earth have any hope of winning the contest of will that would occur in a direct flight straight into the Eye of Sauron, even without considering the draw of the Ring. The strongest good characters in the story are many times described as not certain of their ability to face Sauron's power directly. It's actually a question that comes up a lot, asked about Tom Bombadil, Gandalf, Aragorn, Galadriel, and Denethor, at least. And I don't think all these comparisons are meant to be military, even the talk about Tom Bombadil being overwhelmed as the guardian of the Ring or of Galadriel's contest of wills between her and Dol Guldur. It is not just, or even not primarily, Sauron's military strength that makes him formidable. In the books, his strength is more about force of will & malice & spiritual(?) presence.

Finally, when you add to the eagle flight scenario the Ring, which entices and draws the user according to Sauron's will, it becomes more impossible. Aside from confrontation with Sauron, both Galadriel and Gandalf have hinted that they feel they are not strong enough to be trusted with the Ring. I do not think they would count themselves strong enough to fly directly into the Eye with it unless perhaps they were to claim it for themselves.

Sidenote - Flying Unnoticed

To the argument that they might slip by Sauron's gaze unnoticed, I would first mention that Frodo has to battle against Sauron's gaze/will/power/etc without even being seen. Also I contest that they would likely be seen in the open sky, because this mysterious "spiritual" nature of Sauron's power applies to his sight especially, aside from any Palantir. (He is called the "Great Eye", after all). There are several examples in the books of characters like Gandalf (after the battle of Pelennor fields) or Aragorn (talking about the King's seat above Rauros), having or discussing experiences of seeing without a Palantir more from a high place than could probably be physically, mechanically seen. I'm not a medieval literature scholar but I would not be surprised if this idea was drawn from older romances and fairy tales. Hurin's imprisonment by Morgoth on a high place while he sees the lives of his children is an example of this as well. Considering Frodo's numerous experiences of being hundreds of miles away and feeling the Eye just moments away from seeing him, I think we can safely say that this is part of the nature/power/aura/enchantment of Sauron. How the palantir adds to this I am not sure, but I think it is more than physical sight. Also, the whole attack on the Black Gate was calculated to draw away his attention from his own lands, so his ability to see all that goes on in Morder is clearly a central concern in the story. For this reason, I think it would be especially risky to fly bare and unconcealed into the open sky into Mordor, even high up. There is a large chance they would be spotted, especially given the nature of the Ring. And once they were spotted, they (hobbit & eagle) would be confronted with the strength of all of Sauron's will and malice as alluded to above.

And of course on top of all this are the risks others have mentioned: Sauron's military might, the arrows of orcs and the strength of the Nazgul, who themselves have a terrible power of their own.