I would say it mattered as much in the evolution of cavalry tactics as wing flaps on WWII fighter planes. Sure they're handy on takeoff and during a few advanced maneuvers, but you could easily do without them if you had to. What really made the knight predominant in Europe was the amount of political and economic power available in the knightly heirarchy. In the Classical era the war-winning force was either citizen soldiers (hoplites), free farmers (Republic legion) or professional soldiers (various mercenaries, Imperial legion). However following the collapse of the Roman Empire the trade networks that supported the big cities dissolved, the farmers had all become serfs under landowning patricians, and nobody could afford to maintain standing armies without a broad tax base. The money was with the landowners, however, and since the landowners could keep horses on their estates it was really a natural progression to the knightly system of guarding your land and people by personal military effort, investing in a retinue of trusted men and arming them accordingly, and drawing a feudal levy when the community was threatened. This was not at all unlike the Macedonian Companion cavalry or Roman equestrian class, but the difference was how far they outperformed the infantry. A fyrd or levy could be expected to hold its ground in a phalanx-like mass, but the heavy infantry of antiquity had levels of discipline and tactical suppleness that wouldn't be seen again until the Swiss and German mercenaries. And as infantry evolved into more professional sergeants-at-arms and the nobility relearned the military art themselves, it remained natural to keep themselves as the centerpieces of the chessboard rather than relegate war-winning status to those they governed by the claim of being the principle guardians of their communities.
TL;DR version; claiming stirrups is the reason for the development of the knight is a very tempting way to have a snippet of trivia explain a very complicated issue.