[Saraphim]

I thought the whole arguement against the eagles simply chauffeuring the hobbits to Mt. Doom was the fact that they were trying to sneak into Mordor and destroy the Ring unnoticed, which is hard to do while flying around on giant eagles.

Whether or not the eagles knew about the winged Nazgul is a moot point. If they did not, Sauron and his millions of troops would still pose a serious threat when the eagles try to land on Mt. Doom. If so, then they would have been severly outnumbered by nine Nazgul plus ground troops.