The Eagles are not affected by the tidings of humans, they just don't care. Their pride would keep them from simply delivering people to Mordor like a taxi. The only reason they help Gandalf is because Gwaihir (The Eagle Lord) owes him a favor. But he only owed Gandalf a small favor, not winning the entire war. The main reason the eagles didn't help more is this: The Maia are there to guide. Gandalf is a Maia, he is a teacher to the people of middle earth, and just like as a teacher doesn't take your exams for you, he didn't fight their fights for them. Gandalf was so powerful that he was forbidden from challenging Sauron directly. He could've stood outside the gates of Minas Tirith, farted and every orc there would have dropped dead. Same principle applies to Gwaihir, he can only guide not win the war for them.

Yes. Gandalf's fart is that powerful of a move.

In Hipster terms:
Eagles don't care 'bout us humans and dey ain't taxis. The wizards and eagles are so buff, they could defeat Mordor on their own. Dey like teachers. But dey only dere to guide, not fight. Plus dem eagles are too cool to be taxis.

EDIT: Also, it would've made for one of the worst and most boring books ever.