Even if the
Eagles fly to the easternmost point of the Ered Lithui, that’s almost 500 miles to
Mordor. There’s also a range of
mountains or
hills on the way there, which they would either have to go around, which would be 100 or so miles out of the way, or through them. Then they’d have to try getting past
Barad-dûr undetected. All while on horseback and through enemy territory (also, obviously two small humanoids have a better chance of slipping through Mordor undetected than nine guys on horseback). I ask you, how would this be a better plan than what actually happens, which, I might add, ends with the successful completion of the quest? And, what? There are just going to be friendlies with
horses and supplies waiting at the edge of Mordor? Who would they be, and how would they get there without running into the enemy??
And who would even summon the Eagles? Who even ever does so in the book?
Galadriel and presumably
Radagast, neither of whom is present at the
Council of Elrond. Maybe if either were there, they could have broached the possibility, but they weren’t, so the point is moot and we have to assume Eagles were never an option for anything. Their help at certain points throughout the story is an unexpected blessing.