I would love a comprehensive answer to this question updated post pandemic. I teach high school students, and this is becoming a frequent question, or at least it's paraphrased from questions I've received. Every book I've ever read starts "actual history" (which I know is written, but the little kid in me can't separate how close archeology and history are tied) with Mesopotamia because Cuneiform took off in the city-states, but that was so recent in the span of human history. Africa is the cradle of our species, and I remain unconvinced that somehow it took traveling across an entire continent to develop writing systems, codes and laws, mediums of exchange, etc. Anyway, enough ranting. I'm also very open to anything more I can learn about ancient African history, as well as anything precolonial. Thank you!
u/Commustar talks about sub-Saharan languages a bit here, but this doesn't get to the meat of your question so hopefully more can be said: https://www.reddit.com/r/AskHistorians/comments/7kq0mw/is_it_true_that_there_was_no_written_language_in/?utm_medium=android_app&utm_source=share
I think one issue is that you're overestimating the need for writing to have a functional society. You imply the need for a writing system as a prerequisite for codes, laws, and systems of exchange. You wouldn't make this suggestion of, say, the Vikings, or the Gauls or other continental Celts. Even the Achaemenid Persians, who conquered a huge chunk of the world, didn't write their Old Persian language until later, probably under Darius. The Kassites who ruled Babylonia for 400 years didn't have a writing system of their own. Of course in some of those cases the people they interacted with and/conquered were literate and they were thus written about (like the Vikings) or inserted themselves into existing literary traditions (Kassites and Persians). Africa isn't really an outlier in that sense, no more than pre-Roman Britain, for example. Also it's important to note that sub-Saharan Africa alone is about 100 times larger than Britain - you're not going to find someone who specializes in "Africa."
In human history, even after around 3000 BC, writing and literacy is rare. Independent invention of a true writing system (and the distinction between that and proto writing can complicate the discussion) is extremely rare, such that it only happened a handful of times. We're writing in the Latin script, which traces its roots back to Greece, Phoenicia, and finally ancient Egypt (which is in Africa, incidentally, but I think your question clearly meant sub Saharan Africa). Even places that had a writing system didn't always develop what we would think of as a truly literate society where writing goes above and beyond just a useful administrative tool (I'm thinking of Mycenaean Greece but I don't really know much about that - that's my understanding though).