I am not sure why the "fellowship should have flown to Mordor on eagles" has latched onto popular imagination when it was clearly established that Nazgûl could fly.
I think it was originally (pre-pj rings) a meme in Tolkien fandom bc in the Silmarillion, before Morgoth hatches the winged dragons, the Eagles are like super frequent air rescue feuds ex machines. The fell beasts also arrive in the story after the council of Elrond—they didn’t know at the time
At the time when the Eagles could have taken them, Mordor wasn't yet aware of the hobbits progress or location. It's doubtful (considering the Eagle's speed and numbers) that the Nazgul could have made it back to Mordor in force to stop it. It would have been a pretty quick flight.
the answer, which sounds snobbish, is that no one reads the books. there's like a whole cottage industry on youtube answering lotr questions that are explicitly answered in the books by characters. and i'm not judging, i love those videos, that's just the reason why.
The eagles are also a full intelligent race with speech and their own agendas. The movies treat them like beasts of burden like horses. So I think a lot of people are confused about the eagles thinking you can just jump on one and point it in a direction and go.
Also maybe the eagles didn't want to, they have agency.
The Eagles only obey Manwe, they are not a taxi service
It's maddening because...there's like a reason the characters send Hobbits. They go over it and everything. Sauron is a literal all seeing eye that thinks Hobbits are silly creatures beneath its notice. They provide a blind spot not provided what so ever by eagles
Then fly low. I dont know but walking thru enemy lines for months does not seem a good idea. They could have flown at least over the mountains. It is clear that Tolkien’s chars are not military geniuses.
I had a conversation recently with someone who was convinced that Gandalf's last words of , "Fly, you fools!" meant "Use the eagles," rather than "Don't just stand there watching while i sacrifice myself to buy you time." It's bizarre.
There could have been awesome airial battles.
ALT: a bald eagle is standing in front of a lush green forest