Not necessarily. Lower Nubia was seemingly a historically Cushitic stronghold alongside the Eastern Desert,
Atbay and the eastern coastal plain.
The A-Group, the
C-Group, the
X-Group, and
the Blemmyes all seem to have been some sort of continuity from what I can tell. In fact, as recently as the 6 century CE the Blemmyes (Beja) were in control of much of Lower Nubia having settlements like Qasr Ibrim, Faras and Kalabsha under their control:
Here's a letter from a Blemmye King imparting how they drove
Nobatians out of Talmis (Kalabasha;
their capital) at one point until they were finally defeated by
Silko later on and driven into the eastern desert:
In fact, what I find interesting is that despite how the narrative is that the NS speaking Nubians drove them out of Lower Nubia and into the eastern desert around the end of the classical era, when the Arabs describe the early medieval Beja Kingdoms it seems pretty apparent that Lower Nubia is still predominantly Beja:
Notice how it is the first Beja Kingdom that is where the frontier with non-Muslim lands begins and that it starts just below Aswan. Then notice how the Beja Kingdoms only begin to properly border NS speaking Nubian (Nuba) kingdoms like
Alodia ('Alwa) by the third Beja Kingdom of Bazin while generally centering the NS speaking Nubians more south like
Dongola. It's also pretty clear that the Beja of this time aren't simply desert dwelling pastoral nomads even though, like with their Cushitic kin to the south, that probably would have always been the majority of the populace's occupation:
So it's entirely possible E-Z813 was somewhere along the Nile Valley as well and made its way to Upper Egypt then Lower Egypt and the Levant from there. 2300 BCE isn't even that far back. Ancient Egyptian civilization was nearly a thousand years old by then and
the Kingdom of Kerma already existed whilst the A-Group culture is more than a thousand years older. Kinda weird to think that our male line ancestor may very well have been among some of the earliest
Medjay mentioned and that Agaw-East-South Cushites may have been in what have, in more recent history, been Beja lands.
I'm still skeptical cos it's confusing how the Horn only enters the metal and civilizational age (as far as I know) after 1000 BCE seemingly due to South-Arabian influences but man would it be a doozy if so. I always looked at Sudan/Nubia's history after 3000 BCE as "the history of our close relatives but not ours". Weird to think that at least until around 2000 BCE or so it still might have been "our" history too.