Stirrups were indeed not invented in Alexander’s time, but stirrups are not the cause of the invention of lance-wielding shock cavalry. Late Antique Cataphracts simply had specialized saddles (which Alexander’s companion cavalry lacked) and often acted as charging cavalry.
For TFT purposes, I forget if there are rules regarding saddles and stirrups… but off the top of my head I think you might want some divisions:
Saddle-less cavalry should be significantly easier to pull off the horse. I would adjust the “pull from horse" action by a die in favor of the attacker and make damage of, say, 3 or more require a check to avoid falling (keeping 8 as an automatic fall due to being stunned). Perhaps even make them throw a check influenced by Horsemanship if they are actually involved in a (pole weapon) charge attack while on horseback - hence encouraging them to use ranged weapons or long weapons to jab rather than to charge, which seems the most likely use of companion cavalry historically (with extra-long spears, essentially pikes in TFT terms).
The four-horned saddles used by the Romans or even more securely designed Sassanid saddles - all used without stirrups - allowed the use of a Kontos lance (usually two handed) in charges. I would probably rule these as equivalent to normal mounted figures in TFT in terms of likelihood of falling off. However there does seem to be a slight advantage to stirrups in using one handed lances as the primary mode of delivering the momentum of the charge. As such, perhaps lances should be two-handed for this form of saddle, and using a lance one handed (with a shield) should count them instead as a long spear (as above) rather than as a lance. This could inspire additional modifications - such as the inability to direct horses during the charge. Furthermore the saddles offer stability but they may limit ability to shift the body between the horse’s hexes and/or the options for facing. This could help explain why some continued to ride bareback or with a simple blanket over more stabilizing saddles.
Finally stirrups would follow the normal rules in the book - therefore providing some advantages and particularly flexibility in tactics but not being required for cavalry charges.
Spoiler alert: Don't you hate know-it-alls who moan about neat ideas?
Actually, stirrups weren't invented in Alexander's time. They hit the Levant & Europe after the fall of Rome. Cavalry charges by lancers, as we usually picture them, were impossible before then. I've never heard of Alexandrian cavalry deploying multiple skirmishers, though there is evidence that his cavalry were used like dragoons were (i.e. as mobile light infantry which dismounted to attack the flank & rear of heavy infantry & command units) in the 18th & early 19th centuries.
Considering how small ancient horses were compared to modern ones, I'm not sure how hanging a couple guys off a saddle would be possible, stirrups or not.
Fantasy theme aside, why would anyone who wanted mobile light infantry *not* use carts or chariots? That is, in fact, what those vehicles were used for in the ancient world: as a platform for light infantry wielding missile weapons, and for flanking enemy units.
In any case, before stirrups, a horse's back was far too unstable a platform to permit effective melee or ranged attacks.
See Archer Jones, "The Art of War in the Western World"; John Keegan, "The Mask of Command".