Or is this a poor understanding of African history on my part?
I understand why northern and north-east Africa (Carthage, Egypt, Ethiopia) had large states (because being close to the sea gave them better fertility and access to trade), but what's special about the area where Ghana/Mali/Songhay were? Or is it unrelated to geographical features?
To a degree, I do think it is a misunderstanding of African history.
Central Africa did have large states like the Kongo kingdom, and the Luba and Lunda empires which had deep influence on the surrounding societies in the early modern period.
The Kongo kingdom has received a fair amount of attention from scholars and history buffs, and we have decent written accounts of it from the mid-1400s onward due to contact with Portuguese explorers and missionaries.
The Luba and Lunda empires were further inland, in what are now the Kasai and Katanga provinces of the DRC, and we don't have much in the way of written accounts about them before the mid 1800s.
So, while scholars of Ghana can look to contemporary accounts from al Bakri, and scholars of Mali can read ibn Khaldun's chronicle of the Keita royal family, or read al Umari's account of Mansa Musa's pilgrimage. Scholars of Songhay empire can read the Tarikh es-Sudan by al-Sadi and the Tarikh al-Fattash by Ibn al-Mukhtar. All of these sources are either contemporary descriptions or scholarly works within 100-150 years of the events they discuss. Al Bakri and Al Umari tell some tall tales about just how much gold the kings of Ghana and Mali had, which drew interest in the muslim world and continue to be repeated today. Those are easy hooks to get people interested in Ghana and Mali history.
For the Luba and the Lunda empires, we don't have those sorts of written accounts. Therefore, scholars who focus on central africa (and it is a neglected discipline) have to rely on an entirely different set of tools. Jan Vansina in Kingdoms of the Savannah and Thomas Reefe in the Rainbow and the Kings have used historical linguistics and oral traditions to explore the Luba and Lunda past.
Pierre de Maret has done a good amount of archaeological research in the Upemba depression (the Luba heartland). However, later human activity has disturbed known settlement sites. Therefore, we aren't able to get much detail on aspects of Luba settlement patterns. On the other hand, burial sites are better preserved, and De Maret and others have gained information about the development of social stratification among the Luba from the 8th to 19th century.
So, putting together these archaeological, linguistic, and oral traditions can give us a picture of the past in Central Africa, but there are lots of blank spots in our current knowledge.
So far, scholarly works on the Luba and Lunda haven't found that narrative hook like tales of Ghana's gold or Mansa Musa's gold, that will popularize and get layman interest in Luba and Lunda history.
You also mentioned southern Africa, and the empires there. I am much less knowledgeable about the region, and I hope /u/Khosikulu can chime in here.
Still, I would offer up the states of the Zimbabwe highlands as examples of a succession of organized states in southern Africa between the 11th and 19th century. I am referring to the Mapungubwe kingdom,Zimbabwe kingdom, Mutapa, Butua (?) and Rozwi kingdom.
My limited understanding is that these states exercised control or influence over a smaller area and populations than Mali or Songhay did. Though on the other hand, the Zimbabwe kingdom and Mapungubwe were involved in trade with Swahili and later Portuguese neighbors on the Indian Ocean coast. Additionally, at Great Zimbabwe, the capital of the Zimbabwe kingdom, the population was between 10,000 and 20,000, which is fairly impressive for the 13th-14th century.