This is not a plot hole. The Eagles are the representatives of Manwe in Middle Earth. As mentioned in the link in the accepted answer.
I am surprised that this argument comes up as often as it does, because I am aware of no textual support for this idea. The eagles quite frequently involve themselves in the fight against <PERSON> (or against evil in general):
They rescue <PERSON>, <PERSON>, & Co. from the orcs and wolves. They participate in the Battle of Five Armies. They rescue <PERSON> from Orthanc. They rescue <PERSON> again from Zirak-Zigil. They directly attack the flying Nazgûl during the last battle. They fly into Mordor to rescue <PERSON> and <PERSON>. Given all this heavy involvement, it would be extremely surprising if the Valar specifically prohibited the eagles from flying the Ringbearer into Mordor. <PERSON> nowhere mentions such a prohibition.
The only support I can see for this argument is very indirect: namely, that the eagles are said to be the representatives of Manwë, and in the Third Age, Manwë is maintaining a policy of the Valar not intervening directly in the affairs of Middle-Earth. 88
But the eagles do often intervene in the struggles of Middle-Earth, and there's no indication that they were under some restriction in this case. If the eagles were prohibited from being involved directly in the struggle against Sauron, we might expect that they would have withdrawn to Valinor long ago rather than remain in Middle-Earth.
If you look at the involvement of the Eagles it is very similar to the involvement of the Istari. The Istari are Maia sent to Middle Earth as guides and to provide some help in countering the great powers of those who had fallen. They are not allowed to solve the problems of Middle Earth, only to guide and aid those who are solving those problems. The Eagles play a similar role. Both are forces of the Valar -- who, after the first age that destroyed vast swaths of Middle Earth, vowed never again to interfere directly in the affairs of the mortals of Middle Earth -- and both aid those fighting Sauron. But only where the mortals of Middle Earth could not over come the foe on their own.
The ring is a foe that the mortals of Middle Earth could overcome. And in a way, it is the one foe they must overcome on their own. <PERSON> guides them to this realization and helps them figure out the "how". But he will not defeat the foe itself.
<PERSON> aids them by defeating a peer who had fallen, against which none of them could stand (the <PERSON>) and by helping to counter the movements of one of the Istari who had fallen. The Eagles aid them by providing some mobility and countering forces of the enemy when they take to the air, the realm of Manwe, which mortals are not truly supposed to enter.
All of these actions are consistent with the way the Valar are willing to aid those of Middle Earth. However, flying the ring bearer to Mordor is not consistent with them. That would be direct involvement and solving the one problem that those of Middle Earth need to solve themselves -- their lust for power at the expense of Middle Earth itself. 88