Actually, it could have been the very same horse that he or one of the other riders rode in the beginning. Only 8 horse bodies were found; three were found at once drowned in the water, five more found on the rocks of the rapids below. THis information is in the chapter titled The Ring Goes South. So, one horse could have survived.
I agree that the winged beast would be more formidable to the opponents, so why didn't he use that to begin with? (Apparently PJ thought this also, since he had the Lord of the Nazgul start off from Minas Morgul on the dragon instead of the horse!)
Yes, I imagined that he had his winged beast housed at Osgiliath so he could have pretty quick access to it when he wanted it. I just can't come up with a good enough reason (to convince myself
) why he choose the horse first instead of the dragon!