An easier, less costly way to get the Ring to Oroduin...

[Nibs]

An eagle or two flying toward Mordor would have caused quite a hubbub, I'd think. Sauron would realize something was awry, somehow.

The Eagles were under Manwe himself, and he sends them only when he sees fit. Well, I think we can assume that, since it's not directly mentioned. The Silmarillion didn't mention whether or not the Valar were cognizant of the predicament Middle-Earth was in, but I think they were, especially Manwe.

In other words, I think it makes perfect sense that the Eagles weren't sent... oh, and there would have only been maybe one book. Not cool. Uncool, even.