All of your observations, however astute, depends on the eagles consenting to take Frodo! Hey, we're not talking about a stray MiG here...wait, let's do that again.
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.
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 Nazgūl'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]
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He might have been able to get the Nazgul there first but this would have depended on where the Nazgul where prior to his calling on them.
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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 Nazgūl 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 Nazgūl would be flying over Mordor.