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I don't know if this has been said before, but it's my reason for thinking that they don't have wings:
Lúthien was able to see Gondolin from the air after their rescue from Thangorodrim. Okay, Ancalagon was the first of the winged dragons, and he didn't show up till the War of Wrath. So... if there was some sort of creature who was really capable of long distance flight like the eagles, couldn't Morgoth have used them to find Gondolin? Especially after Hurín brilliantly half gave the location away?
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