Sauron was so corrupted with power that he literally could not comprehend his enemies trying to destroy the One Ring. He was blinded to the possibility that a man could be handed that kind of power and not use it. As of the Battle of the Black Gate, I believe he is under the impression Aragorn has it. The whole point of the Fellowship was to deliver the Ring secretly to Mount Doom to destroy it before Sauron could even recognize his oversight.

Dark forces had a pretty good control of the skies thanks to the Nazgul and Saruman's crow spies, so flying in a beeline to the volcano would have tipped their hand waaay too early. They would never finish the approach if they telegraphed their plan before getting into Mordor.

Also, as others here have said, getting the Eagles to help you in just about any endeavor is pretty difficult anyway. They prefer to remain neutral in regards to the conflicts of mortals. Entrusting them with the Ring directly to just drop it off without the Fellowship was also a no-go since it involved the same danger allowing Gandalf to carry the ring would have - there was a chance that it would corrupt them and they are too powerful to take that chance with. Another evil Maiar (if that was indeed what they were) was the last thing Middle-Earth needed at the end of its Third Age.