Yes it start there; at the micro-family level then cascade up to the national level; this is why development is very hard; not instant coffee thing; because resolving say population takes generation. Kenya is making some progress in that regard; while places like Uganda have not even wrapped their heads around it; Asia is doing well now after taming their population growth.
Having few and well-spaced kids is first step for a family to become prosperous. Every kid you add - you're subdividing your wealth or multiplying(increasing) your poverty.
African remain poor despite economic growth - because chiefly of population growth - not corruption. Even with zero graft - a country will not transform - if population is growing that fast.
All that economic growth is just building more schools, more hospitals, producing more food, to feed, house and clothe more people. Instead of building better schoools, better hospital, better food, to feed , house and clothe the same number of people. There has to be that inflection point. I think it's happen in central kenya - we are not seeing building of say new schools by CDF there - but more better equipping of existing school - but rest of country - it's still pretty much starting new schools and adding new classes to gather for more and more kids.
China big break didn't come from embracing capitalism - but 1 child policy a decade earlier.
Population boom is probably a bigger cause Mwafrika poverty. Anyone with 10 kids they can barely feed is as bad as a corrupt politician.