I know the Eagles have been discussed a lot in this thread, but mostly in regards to their use in LotR. So apologies if this has already been said, but there's another important aspect for not getting the eagles to take them all the way to the Lonely Mountain: it's probably safe to assume that the Eagles would frown on the potential scenario of waking up a dragon: a fire drake is probably the only creature in middle earth that would kick the eagles' asses in battle. Just as they're not eager to tell the elves about their intent, it makes a lot of sense to accept the eagles' help when they need it, but avoid divulging the nature of their quest or even their end destination.
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