You certainly have discussed this concept from a number of angles. I would think, however, that it would have been impossible for an Eagle to carry Frodo toward Mt. Doom without drawing the attention of Sauron's all-seeing eye on top of Barad-dûr. Perhaps Sauron could have destroyed them both. I seems to me that the only way Frodo and Sam could have possibly reached Mt. Doom was by a stealthy approach on foot.