Desert = dry. It doesn’t rain much, but that’s what aqueducts are for. That’s all. They can have lots of plants, and there are big agricultural regions, e.g. Beqaa Valley. Are you just asking for greener ME movies? I’m out of ideas.
One reason why the movies use deserts is because it essentially saves time. The viewer sees it and thinks “Middle East!” without having to explain it. If it is against their desert schema, they may doubt it (I bet this is Southern California!) and also not have that instant “Aha!” moment.
The Middle East is a very large place. There are fertile areas and there are deserts, with the deserts being much larger. The fertile areas tend to run along rivers. The classic Biblical places tend all to be away from the major river valleys. Even Egypt is almost all desert except for the Nile flood areas.
Your question is like asking why westerns don’t take place in wheat fields and forests since most of the U.S. is mostly wheat fields and forests. It’s because westerns are set in the west, not in the rest of the country.
It is probably not true that the major Biblical areas were more fertile 2000 years ago than they are today. Today is not a good comparison at all, because the Israelis have spent the years since 1948 performing massive feats of irrigation.
Overfarming is probably not a major contributor at all in these particular areas. In the first place, farming in Biblical times would be unlike what we think of today in our Midwest or even in the truck farming areas of the northeast. They would be small plots and some general grazing areas. Those can hurt the soil in an area without replenishment - that’s why the Nile floods have been so huge historically - but they’re unlikely to turn areas into deserts. Northern Africa, by which I assume you mean the Sahara, turned from fertile to desert because of purely climatic change and did so long before massive farming could do it in. From Wikipedia:
Israel doesn’t lend itself to oases in the classic Saharan sense, but the fertileness of the lands wherever water is present cuts off quickly. Searching for “the land of milk and honey” brings up this fertile glade, which abruptly turns into rocky hills.
And remember that the Biblical area was two kingdoms, Israel in the north and Judah in the south. Most of the Biblical action takes place in Judah, noticably hotter and drier than the north. It was Israel that was the land of milk and honey, with some green valleys but again scrub hills. I know its counterintuitive that Israel isn’t the ancient home of the Biblical Jews, but the term Jews comes from the land of Judah.
I hate to say this, because movies lie about history with every frame of every movie, but setting Biblical movies in lands that aren’t very green is about as accurate as it gets.
The destruction of irrigation system in Iraq and Iran by the Mongols also brought forth a lot of desertification, turning lots of areas from farmlands to desert. Many of these irrigation systems were the products of millenia of work, some dating back to the dawn of civilisation.