[Ainaserkewen]

It was always my impression that because of the free will of the Eagles, they didn't want to do it. Frodo would have been a mighty burden to ask any of the Eagles to carry. Plus, though Gwaihir(head Eagle) did owe Gandalf that life-debt, carry the Ring to Mordor would have been a hefty favour nevertheless.

I always assumed that the reason they showed up at the Gate was because their Lord and Creator in Valinor commanded them. Then Gandalf convinced them to bring back Frodo and Sam to repay their debt.

The Gods, like Gandalf, couldn't directly use their powers to influence the war. The free peoples needed to fight it themselves. So Eagle Airlines would have been considered direct intervention.