As I thought you might say.
Sorry, but grazing animals have certainly been that numerous in that region as to be capable of modifying vegetative cover. When that happens, the surface emissivity changes enough to induce a surface temperature change of 40°F. That changes the relative humidity and the dew cycle upon which the grasses depended.
It is REMOVING human animal management that is what can cause desertification on a scale that massive, particularly because the entire equilibrium between a human apex predator and its herbivorous prey is disrupted without anything to mitigate the effect. That land had been under continuous management by people for over 9,000 years prior with no problem. Over that time there were solar-related climate events of far greater magnitude than when that desert commenced. Yet at the very nascence of the Nile Valley culture, there was war between the people of the Valley and the people of that grassland, and we know who lost.
With that loss came the loss of those who knew how to steward that land, whether preventing overgrazing grasses or maintaining shrubs, hunting predators, or running the animals to crush locust eggs. The desertification process began almost immediately and progressed from east to west. To this day, Khnum, the creator god of ancient Egypt, lines the road from Karnak to Luxor. He has a ram's head. The pharaoh ruled with a flail and a crook. Yet it was known from the time of Joseph (and now before) that the rulers of the Nile hated nomadic shepherds, for good reason too. The Hyksos kings were exactly such.
Yet at the very nascence of the Nile Valley culture, there was war between the people of the Valley and the people of that grassland, and we know who lost.The wars in the predynastic period (and for most of Egyptian history) were about control of the Nile, which helped bring into being one of the riverine civilizations that lies at the root of all our own histories. The reason nomadic life shrank is because the desertification got worse, not because of the other way around. Even the prehistoric traces of warfare in the area (thousands of stone weapon-heads) were found in the remains of a village which was attacked (or had an internal fight of remarkable proportions) and burned to the ground, and evidently never reoccupied.
There were several major catastrophes over the last ten or twelve thousand years, not least of all the Noachean flood. Something like that could have played a part in turning some of the regions you mention into deserts.