It's true that the natives were initially afraid of horses, but I think that overstates it. You can find some accounts of earlier battles that went like this:

It's true that the natives were initially afraid of horses, but I think that overstates it. You can find some accounts of earlier battles that went like this: > The Indians, who had never seen any horses before, could not think otherwise than that horse and rider were one body. Quite astounded at this to them so novel a sight, they quitted the plain and retreated to a rising ground. [...] Our swords had done the most carnage among them, though many were killed by our cannon. Wherever the cavalry made its appearance the enemy had most work to do. The fighting lasted about an hour; and our enemies maintained their ground so well, that they did not quit the field of battle until our horse broke in among them. There were two caziques among the five prisoners we made. But not much later, the natives recognize the importance of the horses and specifically targeted them: > Besides all this the Indians kept continually throwing sand in our faces to blind us. Here, indeed, the great mercy of God alone could save us. The chief object of the enemy was to capture one of our horses, in which they did not altogether fail; for, as Pedro de Moron on his well-trained mare, attended by three others of our cavalry, was attempting to break through the enemy's ranks, the Indians wrenched the lance out of his hand, and fell furiously upon him with their broad swords, wounding him severely. They gave his mare such a terrific cut with the same weapon in the neck, that the animal instantly fell down dead. This from The Memoirs of the Conquistador Bernal Diaz del Castillo. At any rate, it is perfectly sensible for unmounted infantry to be terrified of horse cavalry. Mounted cavalry is hard for infantry to beat without specialized weapons and tight formations.