By the eighth century a new mode of warfare had emerged (the mounted shock cavalry), which in turn played an important part in creating the institutional structure of both the feudal state and economy. This was dependent on the invention of the stirrup. Before the stirrup, horses were ineffectual in battle because the rider had nothing with which to hold him securely to the horse. Accordingly, a spear could only be delivered through the strength of the rider himself. But the stirrup enabled the rider to deliver a blow with the full strength of the horse. In this way, frail human-muscle power was replaced with superior animal power, enabling shock cavalry simply to plough like a bus through foot soldiers (p.103, emphasis added).Or, in an intriguing online economic history of the world:
The Greeks and Romans also did not use stirrups which allowed cavalry with lances to be used as shock troops in warfare. In antiquity war was mainly conducted on foot, or horses pulled chariots (Chapter 9, p.1).That horses were ineffectual in battle without stirrups, or were incapable of being shock cavalry, would have been news to the Assyrians, Alexander the Great, Hannibal, the Parthians, Palmyrans and Sassanids, all of which used stirrup-less shock cavalry. Mobility and height were sufficient advantages in themselves.
Once horses became large enough across the shoulders to ride, chariots largely disappeared from combat in the main civilisations. This occurred around the time of the Assyrians (who were the transitional case using both chariots and cavalry) about a thousand years before the invention of the stirrup and even longer before its arrival in the Mediterranean world.
Mobility and height were what gave the shock cavalry sufficient advantages to be effective prior to stirrups. Stirrups certainly made such cavalry more effective (though recent recreations of tilting cast doubt on how much) but stirrups were not needed for shock cavalry to be worth having. Indeed, that shock cavalry already existed is important in why the stirrup spread in the way it did, as the type of soldier and trained horses did not have to be created from scratch.
Moreover, stirrups mainly made shock cavalry more effective against other cavalry. That horses will not charge home against steady foot that stood its ground was as true in 400BC as it was in the Anglo-Scottish wars or at Waterloo in 1815. (What horse-and-musket era infantry squares primarily did was get rid of vulnerable flanks and rears.) Horses are pretty pliable, but they aren’t that stupid. The trick for steady infantry was to produce foot that was sufficiently resilient (i.e. that believed their mates would also stand).
The stirrup nonsense gets repeated a lot, but even cursory knowledge of military history indicates that it is nonsense.
But, and this is what really gets me, even cursory knowledge is not apparently sufficiently prevalent. Scholarly folk (including full-grown Professors) read cavalry ineffectual before stirrups but apparently don’t stop to ask hmmm, were there shock cavalry before stirrups? Even if you had never heard of the Assyrians (what, no Bible reading?), Parthians, Sassanids or Palmyrans, surely Alexander the Great and Hannibal should be within one’s (Western) historical consciousness.
Will this nonsense never die? How much evidence is enough?
ADDENDA For Lord of the Rings fans, the charge of Rohirrim at Helm's Deep gets around this problem by having Gandalf light-shock the orcs at the crucial moment. The charge of the Rohirrim against the orc foot besieging Minas Tirith seems to assume the orcs break at the crucial moment, the sort of thing which did happen. It was scary to see a solid line of men and horses coming straight at you with murderous intent. The trick was not standing yourself, it was believing the guys around would stand: if you did not believe that, and broke, you were cactus.