Eagles get the ring. Eagles fly to Mordor. Eagles get shot down because Mordor has the best air defense system in Middle Earth. Sauron has the the ring and throws a Eagle BBQ party to celebrate. The end.
Ongoing diplomacy between Rivendell and the Eagle King about whether the eagles can be used to fly offensively into Mordor or only defensively within "Free Peoples" space, and whether carrying the ring to the Eastern marches of Gondor is proscribed as too provocative.