The mouth of Sauron has spoken

[The Architect Of Fate]

so instead of being there, so that their powers cannot be put to good use lest it be seen, why don’t they just hide where men and elves can’t see them, and send anonymous and seemingly fortuitous, thunders, lightning and tornadoes to get rid of the orcs.

The problem with sword and sorcery fantasy is sorcery would rather tend to make sword superfluous, just as if the power to destroy a planet is insignificant next to the power of the force, then why does the Masters of the Darkside keep building more death stars to be blown up?

It took maybe twenty minutes for the Eagles to fly Sam and Frodo from Mt. Doom to Minas Tirith. If the Eagles flew them from Rivendell to Mt. Doom in the first place the whole trilogy would be skinnier'n the Book of Mormon with all the "exceedinglys" and "it came to passes" yanked out.

If they took the eagles to Mordor they would all be dead as the success of their mission depended on secrecy at not getting the dark lords attention.