Here are some thoughts about the debate of "why didn't they take the Eagles to Mordor?" First, the Eagles' eyries were the wrong direction. You may recall that the eyries of the Great Eagles were to the east of Rivendell, in the middle of the Misty Mountains (which, by the way, were being overrun by orcs and goblins), and Mordor was to the south.
Second, how would they have gotten word of their request for transportation to the Eagles without attracting unwanted attention? You will recall that the only way to reach the eyries was by flying. Trying to get to the Eagles in order to ask them to take some Hobbits to Mordor would have been a perilous journey in and of itself, not to mention they didn't have time for it.
Third, the Eagles are a proud race who help when it suits them (more on that later), and who's to say whether they would have agreed or not?
The Eagles rescued the dwarves and Bilbo from the Orcs because they enjoyed ruining the sport of the orcs. Later, they carried Thorin & Co. to the Carrock in repayment of a debt to Gandalf. At the Battle of Five Armies, I see their assistance as pest control: They were ridding their mountain homes from creatures who were crowding out or destroying their food sources. Sixty years later, when Frodo was leaving Hobbiton, the Eagles had their own problems (orcs were encroaching on their territory again) and wouldn't have been able to help even if they wanted to.
It is unclear to me why they showed up at the Battle of the Black Gate. Perhaps they were not the Great Eagles of the Misty Mountains, but rather the Eagles of the White Mountains (which is a lot closer to Mordor). Or perhaps they started gathering after receiving word from Gandalf. Recall that Gandalf was rescued from Zirakzigil by Gwaihir the Windlord (who, if I remember correctly, was the Chief of the Eagles at this time). By that time, the Fellowship was on the river, being shadowed by winged Nazgul, making it difficult for the Eagles to meet up with them.
Remember, the Fellowship was supposed to be traveling in secret, and that is hard to do when riding on the backs of giant eagles. That is why they couldn't take the Eagles to Mordor.
