Bear in mind that historically chariots in Egypt, Assyria and even Britain were pulled by small horses not suitable for any kind of cavalry- some Egyptian chariots were even pulled by donkeys. The big horses used by modern cavalry up to WWI are pretty recent (as is the stirrup). Even the Huns and Mongols used lots of ponies. Also, a chariot was a mark of nobility and thus an element of battlefield command, and damned expensive to build and maintain, thus a status symbol.
The Casse in the 3rd century BCE could not have trained the heavy cavalry available to the Macedonians: they just didn't have the horses for it. The kind of armies people fought with were a result of the resources they had available to them.