And, yes, I've heard the arguments against it:
- "What about the Ringwraiths? They have flying mounts!"
Sure, but there was a window of opportunity after they were flushed away at the fords that they became a non-issue. They had to make their way back to Mordor, assume new forms, and take up mounts. That would have taken time.
- "Gandalf couldn't just summon the Eagles whenever he pleased. Gwaihir tended to intervene when it suited him."
True. Although the films depict Gandalf summoning Gwaihir via moth (once in The Hobbit and once in The Fellowship of the Ring), the books depict Gwaihir and his fellow Eagles spotting a situation they didn't like and intervening of their own accord. Could Gandalf have convinced him to help? Maybe.
- "Flying to Mordor would be too far."
True enough. After Gwaihir plucks Gandalf from the top of Orthanc, it is mentioned that he cannot carry Gandalf very far, so he takes him only as far as Edoras. But, the journey to Mordor could have been conducted in stages.
- "The book would have been far too short then, and the characters wouldn't have had opportunity for full development."
There we go. The real answer.