I have hear other, more plausible, explanations. Weather is one. The fact that Eurasia is wide means that it has similar conditions in large portions of land, so you can reuse technology for agriculture across all of it.
In Australia and Africa weather is not so good for agriculture. America is rich, but being narrow you need to create new agriculture techniques as you move south or north, as the landscape, weather and other factors are changing.
War is present in all cultures. So some more unique qualities should be looked for. And luck can't be discarded.