No. You're the one who is wrong. There was no mass displacement of people. There was simply the Arabization of those populations. Much of Northern Sudan is considered Arab today. Yet if you look at the people they look black. Many look no different than populations in neighboring countries like Ethiopia and South Sudan. Why are they considered Arabs? Because they adopted the Arabic language and customs. Same thing happened in much of Northern Africa where it was mostly Berbers but they now speak Arabic and are considered Arabs. The same thing happened in the middle east. This Arabization is just not as obvious as it is in a place like Sudan because the Assyrians and Chaldeans for example probably looked very similar to the early Arabs. So when they got Arabized it was not as obvious as when the Sudanese were Arabized.
This is why genetically that whole region is so similar. The majority of Semitic speaking populations outside of Africa carry Y-DNA haplogroup J while the Afro-Asiatic speaking populations within Africa carry Y-DNA haplogroup E. If there was this mass spread of Arabs displacing these native non-Arab populations we would see it in the genetics. But instead what the genetics show is that the Arab world is very distinct genetically with clear separation between the African populations and the non-African populations.