"After Frodo and Sam entered Mordor thru Cirith Ungol, Sauron had fully ten days to contemplate the purpose of their mission into Mordor, and did not realize that their plan was to destroy the Ring until the last moment. Given that the Mouth of Sauron refers to the intruders as "spies", Sauron's most likely thought would be that the eagles are merely flying over his realm for reconnaisance purposes. Thus, even if Sauron notices the eagles, his response would not necessarily be an appropriate one to prevent the Ring from being destroyed (such as having the Nazgûl fly directly to Mt. Doom to intercept the eagles)."
Two spies aren't going to do much inside Mordor, in practice they've somehow got to be able to communicate information outside of Mordor, or get out themselves. Either impossible or very very difficult. In terms of being "spies" they are an annoyance, not a huge threat.
A giant eagle patrolling your skies is the complete opposite, it can see everything, all your forces, where they're going, what they're doing and it can get that information back out again quickly while it's still relevant.
So if you have the Nazgul, why wouldn't you send them after such a powerful creature the moment it enters your territory?
> A giant eagle patrolling your skies is the complete opposite
Not even patrolling. Aimed full speed at Mt Doom in a straight line (according to the proposed map and speed vs stealth hypothesis). It would be reasonable they could be spotted early on, and suddenly the question Sauron has to answer is: "What could a small, fast air force sent directly towards Mt Doom be up to?", to which the answer is obvious. This would make him likely to set up something (interception, barrage...) sufficiently likely to jeopardise the mission. The fact that it is not discussed as a strategy is a much better argument as to why it's a plot hole than any technical point. Still, at that time it is not known what kind of forces stand in Mordor. Besides, the protagonists may not be seeing the task as being so urgent at it looks towards the very end. This makes a prudent approach more reasonable than an all-in decision.
My very personal hunch is that the hobbits (incl. Gollum) are supposed to do this completely by themselves. Many times Gandalf is asked if things will succeed, and many times he worries about their situation and merely hopes they fare well. Still, he refuses to leverage any form of magical or network power to gather any intel — however remote so as not to disclose their location — regarding their progress and simply waits for something to happen. As a plot device, he basically delegates this to Gollum which stands as the guide towards Mt Doom and the Evil that Frodon has to stand against, hence making him stay on the Good side. It is likely that without Gollum, however short the timing (e.g in a supposed fast-flight), Frodon would succumb to the ring.
I never really understood this in the books. How could Sauron not know that their intent was to destroy the ring? It is made clear that Sauron can see Frodo's location when he puts on the ring. Frodo puts on the ring at various points throughout the series, each time increasingly close to Mordor. How could Sauron have not considered this to be their intent?
We understand early on that employing the Ring is too hazardous to contemplate, and will simply lead to a different kind of doom, at best.
Sauron is worried that Saruman or Gandalf or Galadriel or Elrond or Aragorn could successfully command the full powers of the Ring, gather together the might of Men, and beat him down to nothingness, so that he must suffer another 3000 years of powerlessness before trying again.
Sauron may suspect that another could not do as much. But how could he know with certainty? He only has fifth hand reports of Isildur's tenure as the bearer of the Ring. The Ring was intended as a powerful tool for Sauron himself -- its nature in the hands of others may be as much a mystery to Sauron as anyone else.
The drawbacks of the Ring that we become so intimately familiar with over the course of three books are all perks in Sauron's mind.
The approach of the Ring is thus the approach of an adversary, using Sauron's own power against him. Sauron was probably wondering why Aragorn didn't brandish the Ring the moment he arrived at the Black Gate to challenge him.
"After Frodo and Sam entered Mordor thru Cirith Ungol, Sauron had fully ten days to contemplate the purpose of their mission into Mordor, and did not realize that their plan was to destroy the Ring until the last moment. Given that the Mouth of Sauron refers to the intruders as "spies", Sauron's most likely thought would be that the eagles are merely flying over his realm for reconnaisance purposes. Thus, even if Sauron notices the eagles, his response would not necessarily be an appropriate one to prevent the Ring from being destroyed (such as having the Nazgûl fly directly to Mt. Doom to intercept the eagles)."