Why save them?

[Zigûr]

Originally Posted by The Mouth of Sauron

More to the point, why didn't Gandalf and Elrond save everyone a lot of time and trouble and just hand the ring to Gwaihir and let him fly into Mordor and drop the ring into the fire?

Because no one could destroy the Ring voluntarily. It wouldn't have made the slightest bit of difference had they rushed it there on Gwaihir, and besides, in these circumstances without Sauron being distracted, as he was at the Morannon, he would have easily identified the Ringbearer and recaptured the Ring.

The point is: it's impossible to just "drop the ring into the fire." There was no "easy" solution for dealing with the Ring. Let's assume Gwaihir's sent in alone. Being an Eagle, would he have claimed it? He was a sentient being, so I think we have to assume that he would. Even if Middle-earth was a Dungeons and Dragons world where Gandalf could cast some teleportation spell to transfer Bilbo or Frodo or himself or whoever from Bag End to the Sammath Naur instantaneously they still would have refused to "drop the ring into the fire." They would claim it, put it on, reveal themselves to Sauron and be destroyed. The Ring was destroyed by sheer accumulation of circumstances - which is to say, Fate, or the Will of Eru (or maybe by itself operating Frodo's curse on Gollum, or some combination thereon).