Well first off the eagles are an independant force who may not be interested in acting as a taxi service.
Second, the eagles were a proud and powerful species. Puting the ring under the control of any such being was a Bad Idea™, especially when you are travelling at 20,000 feet. If an eagle decides to go Boromir on the way the ringbearer will go splat.
Third, there is no indication that Elrond and Gandalf had any method of contacting the eagles. And the only one who does, Radagast, has disappeared. At the time Sauron knew where the ring was. So there was some value in getting it out of Rivendale as soon as reasonable, rather than waiting on the attempts to contact the eagles.
Fourth, the quest depended on stealth. The whole point was that they were trying to do something that Sauron wouldn’t expect. Once he sees a flight of eagles heading for Mt Doom there will be no denying the fellowship’s mission. By the time the eagles get to Mt Doom they will be facing everything Sauron can muster: fell beasts, thousands of orc archers, nazgul, Sauron himself, plus whatever else he may have lying around. Imagine a dozen or less people and some exhausted eagles trying to fight their way through all that.
Fifth, there was no way the eagles could make it all that way in a single go. The eagles are not beasts of burden. They did carry people short distances at need, the dwarf party a couple dozen miles twice, Gandalf from Isenguard to Edoras, and Gandalf from Moria to Lorien. But we are told that the Eagle that rescues Gandalf can only carry him to Edoras. If that is an accurate protrayal of their range, then they will have to make dozens of stops and need to take days to weeks to complete the journey. Plenty of time for the ring to corrupt an Eagle, Sauron to realize their mission, and/or ambushes to be set.
**This is consistent with the overall strategy, too. A full attack on Mordor will not prevail. It is the secrecy of their quest that gives them their only hope. Also from this cite:
**Neither eagles nor armies (except as a distraction) would have worked against the Dark Lord’s awesome power. Their only hope was stealth.
Sixth, that’s not how it was supposed to be done. In one of their talks before the quest, Gandalf tells Frodo (and I paraphrase from memory) “Bilbo was meant to find the ring, and not by its Maker. And therefore you were meant to have it as well.” Who’s Gandalf talking about? Who meant for the hobbits to be the agents of the ring’s destruction? Well, I can’t see who Gandalf could be referring to except for Eru himself. And if Eru wants the ring destroyed by having hobbits carrying it on foot through Mordor, I submit that giving them rides on eagles is unlikely to go well.
Seventh, supposing somehow the ride on the eagles could have been successful, when Frodo (or some other ringbearer) reached Mt Doom, he would have been unable to destroy it. He couldn’t even toss it into the fireplace in Bag End, for criminy’s sake. No way he voluntarily tosses it into the volcano. So unless someone goes with Frodo to toss him in along with the ring, the airborne quest would be doomed to failure. As Tolkien states on multiple occasions in his letters, the destruction of the ring is ultimately an act of grace (presumably on the part of that Eru fellow again). That grace is granted because of the effort and sacrifice made to get to that point. Quite possibly if Frodo shortcuts to the mountain via eagle taxi service, Eru doesn’t give him a hand with the whole ring destruction deal.
