This would leave the Mid East essentially unrecognizable frankly. We have no idea what was existing within the Southern Levant, Arabia or Egypt linguistically prior and there are not even examples of non Afro-Asiatic languages in these areas form historical record. From what we gather, it is not clear which group predated who within what we term Iraq or lower Mesopotamia. Some say the Sumerians arrived at a later time and subsumed a local culture, which possibly cohabited along with nomadic Semitic populaces. From some name lists in the Sumerian king list, the more ancient names, appear a mixture of Semitic and Sumerian names. One of the kings mentioned that ruled for some obscene period of time (something approximate to 32,000 years) is literally the Akkadian word for scorpion. The area was a general area of constant migration and assimilation of cultures; a situation is thus came to by some thinkers, that within Sumer, there was three different spoken tongues or people, a local Ubaid farming culture/associated language, the Sumerians and the Akkado-Semites.
The upper Levant and Euphrates, it seems that the Semitic peoples are relatively the most dominant populace as early as records are available. However migrations from the north did come, firstly from an Indo-European route and then later from a Hurrian route. Hurrians likewise would likewise spread into northern Mesopotamia across the Tigris river, merging in settlement patterns with the Akado-Semitic populaces. Some examples include Nizu. Despite this knowledge, I do not know the consensus on the Hurrian origins; my instinct would suggest the vicinity of Lake Van and the highlands surrounding it. The relation between the Hatti and Hurrians, is also alluring, if there is one. Gutians and Kassites are a wildcard, along with the Mannaeans, who may be termed people of the Zagros, which possibly formed an extremely diverse collective of linguistic groups due to the separation of communities in remote mountains and frequent ejections of peoples due to competition or conflict. Elam presumably remains existing, all evidences suggest that they are indigenous to at least Susana or Anshan, which one of these two, matters little.
So, the peoples whom you intend to limit in spread, are essentially for all intents and purposes, the indigenous people or language group of most of the Ancient Near East, at least in terms of historical record. The only ones whom we have firm record of and can define as outside of this group that you remove from the tl are:
Sumerians
Elamites
Hurrians/Biai (Urartians)
Indo-Europeans/Hittites/Pala/Luwians/Greeks/Scythians/Cimmerians/Thracians/etc
Minoans
Zagros peoples (Kassites/Gutians/Mannaeans)
Caucasian peoples
Kaska (not sure on their classification, other than not Semitic)
Hattians
And these groups together, inhabit less lands than the group that you have erased from your scenario.