[Kronos]

We're not talking about someone's pet here, but arguably the most majestic creatures in Middle-earth, who are probably Maiarin in origin - I don't know; they may not be Maiar, but that's not my point. The point is these are creatures with their own intellect, their own free will, and they are very well a variable in the "Eagles taking the Ring to Mt. Doom" equation.

The viewpoint that because these are independent creatures, maybe Maiar, they cannot just aid the Ring quest is a nice one but prone to one overwhelming flaw. They aid Gandalf whenever it suits the author. Whenever it is necessary to resolve problems in the plot that cannot otherwise be resolved.

Why did the Eagles save Gandalf, Bilbo and the Dwarves from the Wargs? Was this reason “below” their status as Maiar?

Why did they rescue Gandalf from Orthanc?
Why did they rescue Gandalf from the summit of the mountains?
Why did they rescue the Ringbearer from Mordor?
Why did they intervene against the winged Nazgul?

It appears to me that there are no hard and fast rules about their participation. And nothing to uggest that they could not and indeed would not fly the Ring to Mordor. It appears that they are ok to be Deux ea Machina but just not too much.
The real reason they are not used is because then we would have no book.

About Sauron's preparedness - remember Angband? Thorongil ring a bell? Surely Sauron must remember those flying things and has thought of them when making Mordor. You don't just leave the air unguarded, despite the fact that air power projection during this period is quite nascent - Sauron must take into account that eagles do exist and that they are in opposition to him. He might have a company of archers on the stand-by, waiting to take down anything that does not look like a Nazgûl's winged steed, or some other form of air defense in Mordor, for all we know. [the last sentence was pure speculation, with no actual evidence from the legendarium]

They can fly so high that Archers would be useless. Only at the point of flying to Mount Doom itself would there be danger. The archers would not know which bird carried the Ring and only that one would need to get through. Still seems a vastly more reasonable method than sending 2 hobbits through the Black Gate.

There are few times during Sauron's reoccupation of Mordor when all Nine left - the hunt for the Ring was the last of them. Aside from that, at least one Nazgûl is in Mordor - Sauron only needs a few to man Dol Guldur, and a few to patrol the eastern shores of Anduin. So you can say that at least one Nazgûl would be flying over Mordor.

Given the Eagles willingness to engage the Nazgul, all remaining eight, above the Black Gate I would not expect them to have much of a problem simply holding off one whilst the Eagles carrying the Ring progressed to Mount Doom.