Lack of air superiority for the eagles until the ring was destroyed and the Nazgul along it. If Frodo had tried to fly on one of the eagles, they would have been shot with "black darts" by the Nazgul, mounted on their flying "fell beasts".
"One does not simply walk into Mordor," Boromir says. "Its black gates are guarded by more than just orcs. There is evil there that does not sleep. The great eye is ever watchful."
I'd assume the great eye w/o all the distraction caused by war would've easily sniped the eagles.
The eagles would have easily been seen coming from far away by the Nazgul, the eye, or anyone else.
This is a common question but it doesn’t hold up to scrutiny. The point was to SNEAK INTO Mt Doom and the only reason this was possible was because of the massive losses and distraction Mordor had suffered.
Even better if there is another Eagle in formation with Legelos, ordered to drop the other Eagle and Frodo with an Arrow when they reach right over the Magma in case someone is having second thoughts. Elrond planned the whole thing.