Although I think The Real Findorfin hit on the most important point here (the Eagles are servants of Manwe first and foremost) I also consider control of the airspace the tactical/ strategic key here. The old (WWII) saw is, if you control the airspace you control the war. From Barad-Dur, Sauron had a clear view of his whole domain's airspace; and he had nine Nazgul who flew according to his whim. To send an eagle (or thirty) into that airspace would have been handing the Ring to Sauron on a feathery platter.
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...down to the water to see the elves dance and sing upon the midsummer's eve.
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