According to the definitions given, though it may seem that Eagles do serve as a Deus ex Machina, Tolkien uses them consistently throughout his writings of Middle-earth. If you read Sil, Hobbit and then LotR, by the time you get to the Field of Cormallen chapter , you might be wondering "Where the heck are they already?" They have become expected.
Because of this consistency of use (mostly high or remote tight spots, not just anywhere) I think they don't qualify as Deus ex Machina. Eagles have always been a part of the story.
If Tolkien had used Eagles one time, giant butterflies the next and flying monkeys yet another...then there would be something to talk about.
--Imladrien
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