I think it would be fair to say that early Western/Latin Christianity had a significant gravitational centre in North Africa. I'm not quite convinced that this is the same as being the birthplace.

A few points along the way to that conclusion. Firstly, that North Africa typically excludes Egypt, in terms of the area we're thinking of. We're mainly talking about the area around Libya, Tunisia, Algeria. Centered particularly upon Roman Carthage. Christianity arrived relatively early here, was Latin in its language, and was Rome-independent for the early period. We have significant early Latin texts related to North Africa, including the Acts of the Scillitan Martyrs, one of our earliest martyr accounts and Latin Christian writings; the Passio Perpetuae, both an important early martyr account, and a first-person female narrative; as well as the dominant theological writings of Tertullian and later Cyprian.

Around the time of Cyprian, though, you see a bit of a shift in power relations between Nth. Africa and Rome. And to put that in perspective, you need to keep in mind that in the Empire's East, there were multiple large metropolises, with strong, independent, equivalent lineages and prestige. In the West, only Rome is a truly large metropolis, and only it has a significant historical claim to theological prestige. Which means that the whole swath of the Latin-speaking areas look to Rome for leadership, which becomes more and more formalised as authority over time.

In Cyprian's time (mid 3rd century), the split in Rome between Novatian and Cornelius, as well as Roman writings to Cyprian, illustrates Cyprian's independence from Rome as a centre of authority. But by the 4th century, Carthage is well within the orbit of Roman ecclesial authority.

That said, it continues to be a site of continued Christian theology, not least in being the birthplace and then later residence of Augustine, the single most significant and influential Latin theologian.

I'm not convinced that 'abandoning Christianity after the Arab invasions' counts anything for this thesis. From the mid 5th century, you have the Vandal kingdom in North Africa, displacing and dislocating relations with Nicene Christianity. Similarly, the Donatist church existed throughout this period as a large, at times majority, alternative to mainstream Catholicism. The Arab invasions of the late 7th century, and the de-Christianisation of North Africa, don't detract from it being a cradle for Latin/Western theology and Christiandom.