Breathable airspace is breathable airspace even in Middle-Earth, and Tolkien kept his fantasy world as close to realism as possible. Magic supplements--never replaces--the plausible action of Middle-Earth.
In other words, eagles could never fly higher than Winged Nazgul. Or faster.
Eagles are also not indestructible as the Battle of the Five Armies (both book and movie versions) indicate. They can be killed by ground weaponry (i.e. arrows) and certainly other winged creatures. Plus Sauron's giant great eye has power of its own: call it an anti-missile defense system. In sum, Mordor is fairly immune to flying stuff. The eagles would be spotted immediately.
Of course, this begs another question: Is it believable that Sauron would never suspect that his enemies intend to destroy the ring?
Yes, it is.
When reading about WWII, one becomes aware of how much the Nazis believed in their own untouchability. Note, I wrote, "Nazis," not the German army or, for that matter, the German submarine commanders. The German army and navy were composed of a mix of good and bad and indifferent leaders like in any nation's military. (And many of them despised Hitler.)
It was Hitler--and Hitler's paranoia--that insisted on maintaining constant wireless communication with the German military, a state of affairs that led to the British eventually breaking Enigma. It was Nazi wishful thinking that led to the bizarre and successful career of double-agent Garbo.
|The Crossing by Peter Fiore, a more realistic portrayal|
|than Leutze's famous painting.|
It is easy in hindsight to see the obvious (and I'm sure if Sauron had lived, heads would have literally rolled), but Sauron's weakness is his inability to believe that anyone would actually destroy the ring. Although he captures and tortures Gollum, he misses what Gandalf and Elrond have not: Gollum may be obsessed with the ring, but he lived for generations under the mountains without feeling compelled to do much more with it than catch orcs to eat. Gollum, however corrupted, has the same stamina and indifference to power that make Bilbo and Frodo good bearers.
Sauron isn't totally imperceptive: should Aragorn, Galadriel, Boromir, even Gandalf--any of his "real" rivals--don the ring, they would sooner or later be drawn into the dark. They might momentarily eclipse Sauron (hence his worries about Aragorn); in the long run, however, they would be drawn to his ways: dominion over the lives of others. A military rival is something Sauron dreads yet something he can handle.
Consequently, Sauron reads in Gollum the very thing he sees in others and himself: desire for the ring. He fails to notice the intrinsic toughness that will eventually undo him. Gollum, however ruined by the ring, is hobbit-like enough to eventually care only about wearing it, not wielding it over others. And that indifference to power is something Sauron cannot comprehend.
"The Lord of the Rings--The Book This Time" posts address the following topics: