As the saying goes, "One does not simply ride an eagle into Mordor." Or...something like that.
Anyways, the point is - maybe giant eagles wouldn't have been the best plan for destroying the One Ring of Power. Sure, it would have gotten the Fellowship to Mordor a hell of a lot faster than walking through three separate 2.5 hour (3 hours with the Extended Editions) movies, but once they got there, they'd be pretty much toast right?
We'll remind you here that we had a theory about the giant eagles too, that fits a LITTLE BIT more in-canon with the writings of JRR Tolkien.