Why didn't Frodo use the eagles to fly over Mt. Doom?

[Gorwingel]

The Lord Eagle said in The Hobbit (In a way) that they like to help, if they don't put themselves at risk by doing it.

Well that would make sense...

My personal opinion is that flying Frodo over to Mt. Doom on an Eagle would just not be secretive enough. Weren't the Fell Beasts basically equals of the Eagles because they were created by Morgoth to be a mockery of the Eagles? If there was a fleet of Eagles heading right straight towards Mordor, I think the Nazgul would have noticed it pretty quickly. But then this takes me to another question... Who was more powerful? Were the Fell Beasts more powerful because they had the Nazgul on their backs? Were the Eagles more powerful because they were good, and could think more for themselves than the Fell Beasts? Could the Fell Beasts have easily taken the Eagles down?

Maybe the Eagles were just afraid?