Why not take eagles to Mount Doom? In the book The Two Towers, when Aragorn et al. find Gandalf in Fangorn forest (after they thought he was dead), he explains that Sauron had the power to repel anyone from Mordor if he knew that's where they were taking the ring. Sauron expected the ring was going to Minas Tirith so somebody could use it and become ultra-powerful, therefore his eye and his resources were abroad and not in defense. Therefore if he saw eagles coming he could have repelled them, and figured out there plan and blocked entry to Mordor; but as he couldn't conceive anyone wanting to destroy the ring rather than use it, the hobbits could sneak in unnoticed. Eagles couldn't have snuck in like that and would have been stopped. I explained it better before but don't want to type it all again.