The reason the Fellowship didn't ride the eagles into Mordor was Sauron's air force
In The Silmarillion and The Book of Lost Tales, it is written that Sauron taught the Numenoreans how to "sail against the currents and the wind", which presumably involved steam power. Ironclads and air ships are also mentioned. Now, as we all know, Numenor got destroyed by Eru. But, since Sauron taught them how to make the things, he would still have the ways to make these things in his head! In the appendix for the Lord of the Rings, it states that Aragorn sneaked into Mordor once disguised as someone else. While there, he must have stumbled upon one of Sauron's secret underground airship factories, where he was building a fleet of air ships to unleash on the world once he had completed enough of them. Sometime during the Council of Elrond (we probably didn't see all of it), someone mentioned the eagles, and Aragorn had to shoot the idea down.
So that's what the Numenorean "iron bows" were.
Or, more simply, Sauron had a lot more of those flying critters than the ones the Ringwraiths rode on, and they made Mordor's airspace unattackable until Sauron's fall. That's why Gandalf could ride one to save Frodo and Sam afterwards — the creatures had fled in fear after Sauron fell.
Here's a simpler explanation: it wouldn't have worked because Sauron could control the air itself. See the Caradhras chapter.
It's implied that Caradhras itself is attacking the Fellowship, not Sauron, although nothing's definitely stated...
I thought the same thing as the above troper, the fell beasts would probably have eaten them or caught them and done something much, much worse.
Someone actually sent a letter to Tolkien asking why the Fellowship haven't flown to Mordor. He got so angry he draw the logo of the Mordor Air Defense and sent it in the reply. True story.