The entire point of the books is missed if the eagles drop the ring in. Sorry, not a great ring of power, just a magic bauble of the sorts you find in D&D. It's not just a story of a great big adventure. The point is the struggle with the ring, the struggle between the factions, the struggle most of the people had with themselves, and so forth.
Or to use a word from Pratchett: narrativium.
The Hamlet Defense! (I.e. why didn't Hamlet just kill his Uncle in Act I?)
Very true and an entirely legitimate narrative device; it's just much more impressive when the storyteller shows awareness of the hole (and preferably has a bit of fun with it). Like if you had that last host of the Captains of the West surrounded by all the armies of Morder at the Black Gate, and then the eagles show up, and Merry turns to Gandalf and is all "Er... Why didn't we--"