Look at it like this. By the time Gandalf found out it really was the one ring the nazgùl had already left Minas Morgul and were moving towards the Shire. Gandalf made a wrong choice and went to see Saruman. He got captured and when finally he got found by Gwaihir he had no idea of where Frodo was and the rangers had more chance of finding him than the eagles. Then Frodo, the other hobbits and Aragorn got to Rivendell. Here they would have had the choice to call the eagles and ask them to take the ring to Mount Doom. However by the time the eagles would have gotten there the Nazgùl and Sauron were already watching the borders of Mordor and it would have been impossible for any creature as large as an eagle to bypass that vigilance. Not even a hobbit would have been able to pass if the whole vigilance on the borders of Mordor remained.
If I remember well, it is said in the book that Sauron's downfall was the fact that he was prodded in to action too soon. Had he just remained in Mordor and bided his time he would have amassed an undefeatable army and nothing would have been able to pass his borders. Once the ring had passed within Mordor in the hands of Frodo and Sam the eagles were out of contact since they were in the Misty Mountains and besides it would have taken them quite some time to get to Mordor. Theoretically they would have been able to get to Mount Doom, but I think that while the Nazgul flew on wings they had too small a chance.
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