These areas were, indeed, fertile for thousands of years before human writing existed. The encroachment of the desert over the centuries has covered much of the area, and it's expected to continue expanding south into Africa.

Your assumption is completely off. The places you mention aren't deserts. They are the opposite. The area is known as the fertile crescent as the local geography and large rivers allowed for early settlements. There's plenty of desserts around these places though.

The people of Mesopotamia, Egypt, Persia, Elam, Assyria, and Babylon where able to thrive as empires before the people of Greece simply because they had more fertile land to work with. They had more fertile land than the Greeks and the Romans because of the three agriculturally helpful rivers the Tigris, Euphrates, and the Nile. The Greeks and the Romans possessed no rivers like these in their lands. These were rivers that had very fertile shores and since the rivers were huge, it allowed for the people of the Ancient Near East to make an ancient form of irrigation in their lands. They did so by selectively flooding plots of land using the many canals that were built in the lands of Akkad and Sumer (more being built in Sumer near Lagesh, Ur, Babylon, Uruk, Kish, and Emma than near Akkad and Nippur). The mighty rivers supplying these canals were essential for the ancient Mesopotamian cities’ survival and growing their barley, cereals, and wheat. This has proven to be the case this why the city of Eridu was eventually abandoned because the Euphrates River had actually moved and meandered away from the ancient canals of Eridu and made the city inhabitable. Additionally, the Assyrian empire in Northern Mesopotamia were able to thrive even though they were at the top of the Tigris River because of the heavy rainfall. Ancient Assyria was able to grow crops because the rainfall was enough to support Assyria’s people with water and the crops they could grow (although it should be known when Assyria had power during the Neo-Assyrian Empire, they controlled much of Mesopotamia and its agricultural resources so this didn’t matter). So really the Mesopotamians were able to develop craft specialization in the arid climates they lived in because they learned how to use irrigation properly in the lowlands between the rivers.

Egypt was also able to unitize the water of the Nile just as well because of the yearly flood they had. This yearly flood happened just after they harvested and stored the rice and other grains they grew and prepped the land each year. This basically flooded the land the same way the ancient Mesopotamians did but it was just a natural effect of the Nile River rather than man made. This allowed for amazing crop yields in Egypt that made it one of the main breadbaskets of the ancient world until the fall of the Western Roman Empire. So it was only because of the proper use of the massive amounts of water coming from large mountain ranges like the Caucus, Tarsus, and Zagros Mountains that the people of the Ancient Near East and Egypt were able to develop craft specialization so easily. This was something that did not happen in Ancient Greece and Roman before Rome conquered the known world solely because of their geography.

Like it was said before, the cities and places were the ancient Greeks and Romans settled did not have the luxury of a huge, flat expanse that had a huge river running through it. The lands of Europa where the Greeks and Romans lived were very hilly and mountainous and not easy to cultivate (though these mountains were much smaller than ones in the Ancient near East). Besides growing the occasional wild grain or wine on the hills, there was very select agricultural land that was like the irrigated lands of Mesopotamia. Besides the Po Valley where the Romans got lucky and the Eurotas Valley where the Spartans got lucky with two rivers (since they conquered Messene), most of the land was nothing compared to the fertility of the Mesopotamian land. So even though the Greeks and the Romans used their favorite rivers like the Tiber and all the corresponding rivers near each Greek poleis, they just weren’t as fertile as the Tigris, Euphrates, and the Nile were. So among the many drought, possible earthquake, and geographical issues amongst the Greeks and limited space and constant cultural conflict of the young Roman culture, they weren’t able to thrive as fast because they were not able to unitize a river as much or as effectively as the ancient Mesopotamians or Egyptians could.

/r/askanthropology is probably a better source, but I feel I can give you a pretty good answer from coursework and readings about the comparative development of societies in agricultural vs pastoral parts of asia.

Part of the reason is that while those regions you mentioned are largely arid by land-area, they contain some of the most fertile farmland on the planet, the Nile delta being a prime example.

The second is that farming in these areas requires more societal organization than in a more fertile area. Agriculture in much of the Iranian plateau, mesopotamia, etc, relied upon an extensive network of canals to make the best use of the water that was available. This leads to a combination of high population densities (extremely fertile soil near rivers) and the need to develop more structured and technology-oriented societies than steppe nomads or small-hold farmers with less need for irrigation.