If Gandalf were paying attention, he would have realized immediately that Bilbo's Ring was the One. While Sauron was retreating from Dol Guldur, Gandalf would have, using Shadowfax's sire, raced him to Mordor. At that point it would be a sprint for the Sammath Naur. Gandalf, having legs, could easily outrun the Eye.
Sauron would be obviously motivated; Gandalf would be likewise, being foresighted, knowing that if he didn't destroy Sauron then and there, he would be cursed later to journey the better part of a year with Pippin.
Run, Gandalf, run!
Another strategy would be for the Eagles to take Bilbo aside and simply drop him into Mount Doom.
Sure, you'd still have fire and ruin in the north, and a Balrog loose in Moria, but it's not up to us to right all wrongs, but to give others clean earth to till (or something).