I think Freeman Dyson may have a point, but his facts seem to lack foundation. One only needs hay for horses that lack sufficient winter pasture for forage.
Cattle were domesticated by 6,000 BC, and horses by 4,000 BC. Horses are able to forage during winter by using their hoofs to paw through ice and snow, and to break ice to get water. Unless the weather is very severe horses do not require winter fodder if they can forage. This is discussed in The Horse, the Wheel, and Language:
How Bronze-Age Riders from the Eurasian Steppes Shaped the Modern World.
Most cattle are not able to fare so well, as they tend to push the snow aside with their noses; once ice forms they are often not able to feed themselves or drink. Thus one must bring cattle in, and feed them fodder. Today's hay often consists of alfalfa, clover, and timothy grass; it also requires careful drying if it is not to become moldy. Other types of fodder include turnips - our neighbors fed turnips to their turkeys - and other vegetable matter. The Romans used turnips, according to Pliny
When did farmers start "making hay" for winter fodder? Making Hay in Colonial Pennsylvania describes recent times, and why it was required. The history of forage preservation mentions evidence from ancient Egypt, from the Middle East, circa 700 BC, and early Rome, where it was recorded that German tribes, the "Teutons", buried green forage in a pit, covered with dung. The methods discussed are called ensiling, methods for creating silage.
Hay is different from silage because it is dried. The article assumes that making hay is just as old as making silage - because it is simpler. All you do is cut the hay, and let it dry in the sun. Making hay in Germany, which lacks sun, but not rain! makes silage more important. Perhaps the hay rick was invented in Germany, during the middle ages.
Loose stacked hay built around a central pole, supported by side poles in Romania