this map better describes the three regions, and the areas they controlled at the end of antiquity, before the muslim invasion north.
it's easier to see what happened by consulting this image.
when the romans and persians were done killing each other, the saracens and axumites moved in and established a new religion in the elite. but they were quickly absorbed into the local population.
this was a period of chaos and anarchy. in addition to the heraclean wars, there was an outbreak of plague that devastated the population and a breakdown in trade that was probably the result of the ethiopians controlling all of the red sea ports for the first time, blocking roman access to the indian ocean and persian access to the mediterranean. what the archaeology shows is that the middle east underwent a period of massive depopulation from about 600-800, not that it underwent a period of invasion or replacement. the cities of antiquity in the levant were overgrown with weeds. the fields reclaimed the land.
when the dust cleared, people wanted to know what happened, so they made up this history in the 9th century to explain what happened. the truth is that the people alive in the 9th and 10th centuries, writing this stuff in syria and iraq, had no idea what actually happened, and no way to figure it out. so they made it up.