Wait... do holes that meet up with other holes count as separate holes? The nostrils meet up with the mouth hole but then combined they go through to the anus. Do those count as three distinct holes, since looking at each of them individually they would count as a hole? Or would there be one hole with... four openings?
ETA: I have more questions. Do all the openings need to be open to still be considered a hole? Is it only a hole if all my sphincters are open and my mouth is open? If I'm getting choked but my mouth is open and I'm trying to breathe, does my mouth to nostrils count as a hole themselves? Do my nostrils automatically count as a hole as long as one or both isn't stuffed up?
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.