The thing about the eagles would have worked because they could have distracted Sauron long enough to fly into Mordor and drop the ring, just like they do in that video of How it should have Ended
I don't think that would work either. Sauron was tuned to the ring. If they started flying in, I think he would have immediately sensed them coming. It would be a much more dangerous approach and much more risky, with the fate of the world at stake. Who knows if the eagles would even have done it. Of course, it is a fictional work, but it is fun to speculate.