The Afro-Asiatic Uhremeit was probably ancient Egypt. Arabia and the Middle East were originally inhabited by non-Afro-Asiatic speakers particularly the Hurrians in upper Mesopotamia and the Sumerians and Elamites in Arabia and lower Mesopotamia. Afro-Asiatic languages fan out from Egypt in all directions (Semitic languages in the east, Cushitic languages in the south, Amazigh languages in the west, Hausa in the southwest, Egyptian languages in the center) which makes logical sense, whereas an Arabian or Syrian origin doesn't, or there would be more eastern and northern Afro-Asiatic language families if that were the case.
The Semitic historical documents (such as they are, clay tablets and the like) even say this, that Semitic people travelled out of Egypt into the Middle East. The Akkadians write how they migrated to Mesopotamia and learned civilization from the Sumerians. The expansion is probably related to the Green Sahara period about 10,000 years ago where the Sahara desert became a grassland and people confined to the Nile Valley would have struck out for new lands.