Let's not downplay the stirrup too much. The stirrup, with an early toe version in India around 200 BC but the modern version first appearing in Jin dynasty China around the 4th century AD, provides a rider with great leverage and support. The stirrup alongside successively larger versions of the couched lance and gradual improvements to heavy armor from mail to full plate led to the devastating success of mounted European knights from the High to Late Middle Ages, at least until the improvement of gunpowder weaponry in the Early Modern era rendered them less effective.
You're right, though, that shock cavalry existed well before then. The Macedonian Hetairoi Companion Cavalry were undoubtedly shock cavalry able to land decisive killer blows from the flanks and rear of enemy armies in the classic "hammer and anvil" tactic when combined with phalanx infantry. Also, early versions of the saddle in a padded treeless form existed among ancient Scythians, Assyrians, Armenians, Greeks, Macedonians, etc. Perhaps without sharing such knowledge with each other, the ancient Han dynasty Chinese and Romans seem to have both separately invented their own four-horned saddles by the 1st century BC, within the EBII timeframe! While these saddles lacked stirrups, they nevertheless provided decent support for heavy cavalry charges.
To answer the question of the OP, I usually use 4 to 6 cavalry units in a full stack army regardless of the faction or culture. I also had no qualms using Black Sea horse archers as the Romans, the Pergamon and Koinon Hellenon Greeks, the Macedonians, or even the Carthaginians. If you expand that far as an empire you might as well take advantage of local units and fighting styles.