Jerry Pournelle frequently comments that the invention of the horse collar (https://en.wikipedia.org/wiki/Horse_collar) was one of the most consequential ones in history. Here we have a similar move from oxen to something more efficient.
That reminds me that Egyptians and Romans rode in chariots rather than on horseback because they didn't have stirrups:
http://en.wikipedia.org/wiki/Stirrup
The Roman legions used saddled cavalry. Chariots were only really used for racing. But yes, Romans rode without stirrups.