I mostly agree with you for a part Andy, The terme A MAZIGH (meaning free people in Berber) designates all the A MAZIGH nations in North Africa that use the same language, but they have not specificaly the same origines or ethnicity. Some are caucasoid and the southern sub-saharian tribes much closer to Black african borders are negroid type. but more you go towards the north more you meet the caucasoid population.
If we listen to the historians Carthage was desert before the in -trusion of Elissa Didon, fleeing the opression of her brother Pygmalion king of Tyre (actual lebanon)
Elissa Didon fled to Cyprus first with a handfull of her fellows and from there on to North Africa with some others Cypriots.
Carthage was already peopled by A MAZIGH people ruled by king HIARBAS. King HIARBAS received Elissa and her fellows friendly. Because of course they were just a few and he did not ses in them any danger.
Elissa Didon did not move the whole population of Phoenicia and settled it in North Africa!!
Carthage was founded on an A MAZIGH territory!!