Alright, you little whores. It's perfectly simple.
The Ring of Power corrupts everyone, and the more powerful you are, the greater its allure. Which is why big, brawny men like Boromir tend to go mad simply by being in its vicinity, while demure and unambitious creatures such as the hobbits are able to resist a little while longer, but will in time succumb to it as well.
The eagles are powerful demigods of the same rank as Gandalf, who incidentally refused to bear the ring for the very fear of being seduced to keep it for himself and carve out his own personal realm of terror. And that's coming from a guy whose patron deity was the literal Vala of mercy and compassion. Imagine what the ring would do to the minds of semi-feral terror birds accustomed to preying on farmers' cattle and carrying them off to their lairs and eating them whole, and who didn't really like men to begin with.
The whole "fly the eagles into Mordor" idea is retarded bullshit on these grounds alone. The eagles would have dropped Frodo on some sharp rock mid-flight and kept the ring for themselves 5 minutes after departure.
So we don't even have to get into the whole "how many MANPADS did Sauron allocate to his frontline anti-air units" dumbfuckery. The "plothole" isn't "solved", it downright never existed in the first place.
Inb4
hurrrr but Tolkien only thought about this later in the appendices
No. It's right in the very first edition of his book. Re-read the council of Elrond chapter and then kys.