Also, not having information is not a plot hole. We don’t know why they didn’t, and it’s safe to say that considering how intelligent Gandalf is, he would have thought of it, and found a reason it wasn’t feasible (where sending the eagles after the Eye falls was the best choice).

We can speculate various reasons it might not work (The Eye would be able to spot them and archers shoot them down, The One Ring might be too tempting to the eagles, giant eagles lack the dexterity to keep track of a ring, or eagles are willing to make a contribution for a moment, but aren’t willing to really bear the burden and seriously risk their lives), but it’s completely irrelevant. We don’t need to know why. Just because we weren’t offered a reason, doesn’t mean there wasn’t one.