If Frodo were to fly to Mordor on an Eagle, there would have to be an armada of Eagles to protect them from the fliyng Nazgul. Many Eagles would have to die protecting Frodo and the one who carries him. The eagles were most likely not willing to do this. Also there is a chance that the Nazul would have killed frodo, or the eagle that bore him and all would be lost. It is better to sneak in rather than an all out attack. I think the Council of Elrond chose the smarter thing and that's why it worked.
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'They say that the One will himself enter into Arda, and heal Men and all the Marring from the beginning to the end."
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