OH FOR GOD'S SAKE CAN WE STOP WITH THE EAGLES ARGUMENT? Yes, I realize it appears to be a plot hole to people who have little understanding outside of the one book/movies. However, it is not.
They couldn't just fly to Mordor for reasons that are established, very well established, but not through The Lord of the Rings. The eagles, as they are in Middle Earth, are established in The Silmarillion. They are not one of the free "races of Middle Earth" which would include men, elves, and dwarves. Nor are they one of the corrupted races, like orcs are trolls. They are one of the Valar's watchdog races, more akin to the Ents than anything else. Specifically, they are Manwe's own watcher race, whereas the Ent's exist due to the influence of Yavanna. As far as it goes, they are generally completely disconnected from the fate of Middle Earth and the races that live there, and answer only to the Valar, and then only in a questionably direct way. (The eagles seem to be far more directly in touch with Manwe than the ents are to Yavanna)
This does not mean that they do not or cannot act, but they do not generally do so on the argument or request of the mortal races, or even the firstborn elves. This is readily apparent in the sheer amount of time and debate it took for the ents to decide to attack Orthanc. However, the eagles do act and somewhat frequently. They are instrumental in the the War of Wrath at the end of the first age, they appear during the Battle of Five Armies, they save Thorin's Company from the orcs and finally appear at the Black Gate to challenge the Nazgul. However, they do not generally do so at the request of the free races. Their behavior can be more viewed as an act of the gods than their own will and interest in Middle Earth. The one real exception to this is Gwaihir (eagle leader during the Third Age) acting twice to specifically save Gandalf (a Maia, demi-god) as an act of reciprocity for Gandalf saving his own life.
One may simply "walk into Mordor" but one does not simply "ask the eagles for a taxi," and expect a positive result. Doing so would be akin to asking the gods to solve your problems for you. The appearance of the eagles at any given point is more a boon of Manwe for taking the right action than anything else.
Case in point, note the number of times that the eagles intervene in the Third Age vs the First Age, which was a far more tumultuous time. In the Third Age, I mentioned, above, three separate appearances in less than a century, whereas in the First Age there are only two really noteworthy examples of direct intervention (Thorondor saving Fingolfin's body and permanently marring Melkor's face, and Thorondor directly assisting Earendil with the fight against Ancalagon the Black in the War of Wrath). While the First Age appears to be a fulcrum period on a much larger scale, the actions of the few in the Third Age stand much more on the edge of the knife and their actions are far more dire. The presence of the eagles gives greater aid in this period and is a greater blessing.
Basically, to boil it down, Tolkien reserved the eagles as Manwe's trump card, to be played sparingly and only when it was of dire importance. The people of Middle Earth know this, Gandalf knows this, and it doesn't make it a plot hole if you don't know this because your understanding of the lore is limited. It's an established thing. You just have to delve deeper.
They couldn't just fly to Mordor for reasons that are established, very well established, but not through The Lord of the Rings. The eagles, as they are in Middle Earth, are established in The Silmarillion. They are not one of the free "races of Middle Earth" which would include men, elves, and dwarves. Nor are they one of the corrupted races, like orcs are trolls. They are one of the Valar's watchdog races, more akin to the Ents than anything else. Specifically, they are Manwe's own watcher race, whereas the Ent's exist due to the influence of Yavanna. As far as it goes, they are generally completely disconnected from the fate of Middle Earth and the races that live there, and answer only to the Valar, and then only in a questionably direct way. (The eagles seem to be far more directly in touch with Manwe than the ents are to Yavanna)
This does not mean that they do not or cannot act, but they do not generally do so on the argument or request of the mortal races, or even the firstborn elves. This is readily apparent in the sheer amount of time and debate it took for the ents to decide to attack Orthanc. However, the eagles do act and somewhat frequently. They are instrumental in the the War of Wrath at the end of the first age, they appear during the Battle of Five Armies, they save Thorin's Company from the orcs and finally appear at the Black Gate to challenge the Nazgul. However, they do not generally do so at the request of the free races. Their behavior can be more viewed as an act of the gods than their own will and interest in Middle Earth. The one real exception to this is Gwaihir (eagle leader during the Third Age) acting twice to specifically save Gandalf (a Maia, demi-god) as an act of reciprocity for Gandalf saving his own life.
One may simply "walk into Mordor" but one does not simply "ask the eagles for a taxi," and expect a positive result. Doing so would be akin to asking the gods to solve your problems for you. The appearance of the eagles at any given point is more a boon of Manwe for taking the right action than anything else.
Case in point, note the number of times that the eagles intervene in the Third Age vs the First Age, which was a far more tumultuous time. In the Third Age, I mentioned, above, three separate appearances in less than a century, whereas in the First Age there are only two really noteworthy examples of direct intervention (Thorondor saving Fingolfin's body and permanently marring Melkor's face, and Thorondor directly assisting Earendil with the fight against Ancalagon the Black in the War of Wrath). While the First Age appears to be a fulcrum period on a much larger scale, the actions of the few in the Third Age stand much more on the edge of the knife and their actions are far more dire. The presence of the eagles gives greater aid in this period and is a greater blessing.
Basically, to boil it down, Tolkien reserved the eagles as Manwe's trump card, to be played sparingly and only when it was of dire importance. The people of Middle Earth know this, Gandalf knows this, and it doesn't make it a plot hole if you don't know this because your understanding of the lore is limited. It's an established thing. You just have to delve deeper.