My understanding is that humans, as we know them now had moved deep into Europe and Asia by 20,000 years ago. Was this just luck of the draw or were there other contributing factors to large kindoms arising out of the deserts?
Civilizations didn't emerge out of the deserts. The Middle East is home to a very fertile - if volatile - river valley - Mesopotamia, birthplace of Akkad, Sumer, Babylon, etc.
It should be mentioned that the Middle East wasn't exactly the first place where this happened. Egypt (as far as I know) was established some time earlier, also along a fertile and much more reliable river.
You should absolutely check out this old but fantastic answer. Essentially there is a real important reason we call that region the Fertile Crescent. It doesn't really deal directly with your question about why civilizations emerged there, but it might fill in some gaps.
Why is the Fertile Crescent now desert?
Sadly the best answer is from a now deleted user, but you can still read it and all the responses.