Why doesn’t Frodo just ride an eagle to Mount Doom? Erstens: Was wäre das für eine langweilige Geschichte? Zweitens:
Why does Elrond send Nine Walkers, without steeds? Because Sauron and Saruman are both on the lookout for anyone who might have the One Ring. They are both using all the magic and manpower (orcpower?) they possess. All the roads are being watched. Some of the birds are spies.
Riding horses would just bring attention to the Fellowship. And riding Eagles? Why not just construct an enormous neon sign that says in all-caps “HEY SAURON – THE RING IS RIGHT HERE.”