8. "The Valar would prohibit the eagles from such a direct intervention against Sauron."I am surprised that this argument comes up as often as it does, because I am aware of no textual support for this idea. The eagles quite frequently involve themselves in the fight against Sauron (or against evil in general):
They rescue Bilbo, Thorin, & Co. from the orcs and wolves.
They participate in the Battle of Five Armies.
They rescue Gandalf from Orthanc.
They rescue Gandalf again from Zirak-Zigil.
They directly attack the flying Nazgûl during the last battle.
They fly into Mordor to rescue Frodo and Sam.
Given all this heavy involvement, it would be extremely surprising if the Valar specifically prohibited the eagles from flying the Ringbearer into Mordor. Tolkien nowhere mentions such a prohibition.
The only support I can see for this argument is very indirect: namely, that the eagles are said to be the representatives of Manwë, and in the Third Age, Manwë is maintaining a policy of the Valar not intervening directly in the affairs of Middle-Earth. But the eagles do often intervene in the struggles of Middle-Earth, and there's no indication that they were under some restriction in this case. If the eagles were prohibited from being involved directly in the struggle against Sauron, we might expect that they would have withdrawn to Valinor long ago rather than remain in Middle-Earth.
This is the actual reason, but there is more evidence to support it than the OP gives, but like most things Tolkien, you need to read outside the usual narrative to find it. The best explanation for this is how the rules of Manwe govern the wizards, particularly Gandalf:
Emissaries they were from Lords of the West, the Valar, who still took counsel for the governance of Middle-earth, and when the shadow of Sauron began first to stir again took this means of resisting him. For with the consent of Eru they sent members of their own high order, but clad in bodies of as of Men, real and not feigned, but subject to the fears and pains and weariness of earth, able to hunger and thirst and be slain; though because of their noble spirits they did not die, and aged only by the cares and labours of many long years. And this the Valar did, desiring to amend the errors of old, especially that they had attempted to guard and seclude the Eldar by their own might and glory fully revealed; whereas now their emissaries were forbidden to reveal themselves in forms of majesty, or to seek to rule the wills of Men and Elves by open display of power, but coming in shapes weak and humble were bidden to advise and persuade Men and Elves to good, and to seek to unite in love and understanding all those whom Sauron, should he come again, would endeavour to dominate and corrupt. Unfinished Tales, Ballantine paperback, p. 406.
The Valar are allowed to send the the wizards to Middle Earth with the consent of Eru, with constraints on what the wizards are allowed to do. This implies that Eru, the overarching God of Tolkien's universe, has a plan for events in Middle Earth, and that he doesn't want the higher powers (the Valar) to interfere directly.
The eagles are also servants of the Valar, and are not permitted to directly interfere with events. Discounting the events of The Hobbit, the eagles only show up to help when everyone has already made the right decision informed by faith and hope. When the eagle finds Gandalf on Zirak-Zigil, he had just sacrificed himself to defeat the balrog and he had been returned to Middle Earth (clear Deus ex Machina there too). When they attack the Nazgul in the last battle, it was after Aragorn and the other leaders had committed their armies to a "hopeless" last stand. When they flew into Mordor to rescue Frodo and Sam, it was after they had finished the quest.
There are multiple times where the intervention of Eru in the events of Middle Earth are implied. Gandalf hints that Bilbo was meant to find the ring. Gollum steals the ring and falls into the fire. A wind out of the West blows away the smoke cloud that appears when Barad-Dur crumbles (and the smaller smoke cloud that appears when Saruman is killed). So the prohibition against direct Valar action in Middle Earth is no small thing.
Would it have been nice if the eagles were discussed more in the book? Probably, but it was pretty long already.
If you are interested in this sort of stuff, I recommend listening to the Tolkien Professor podcast.