No, good horses need proper grain, not merely grass. One of the advantages steppe ponies have is that they can subsist on grass alone, where most horses need some kind of grain to supplement their diet, or their health deteriorates.

You're also failing to account for climate change over long periods of time, we've had warm and cool periods over the two and a half millenia since the Hellenistic era, and that's also a long enough span for rivers to change course/appear/disappear.