Yes, I never even considered that as an option until I read someone asking that question because a Big Thing is made of how it takes two little tiny hobbits only and Minas Ithil being emptied by almost all the garrison having been sent to the Pelennor and the Captains of the West keeping Sauron distracted at the critical moment on the north gate for the ring to get through.

Giant Eagles would not do it, whether the Nazgul were on winged beasts yet or not. I am sure Sauron would have managed some sort of anti-aircraft defense. If they could train a moth, now...