It did not happen in Africa.
In North America, megafauna (dire wolf, saber-tooth cat, cheetah, horse, camel, mammoth, mastodon, giant sloth, 30+ genera) were killed off not by human activity but by a comet strike c.10818 BCE. It did in the Clovis culture, too. There were simultaneous extinctions in South America and Africa.
For many decades anthropologists and archaeologists earnestly insisted the Americas were barely populated, pre-Columbus. Now we have found major earthworks projects in both North and South America that required societies in, at minimum, tens of millions to have built and maintained.
We know appallingly little about the recent history of our species beyond ~6000 ya, with only tiny, isolated digs at older sites.
A major earthwork in Java, Gunung Padang (previously assumed to be a natural feature because of it size), dated to 22 kya has had excavations halted since 2014.
I would hardly be surprised if rural population density was significantly higher than in the past, with more farms as well as more grazing herds, even if they were much smaller than what we have today.
But I also don't really know if this squares with the sheer amount of meat and dairy that modern humans consume. How much does factory farming offset the need for open grazing land?
Basically, this is a subject area where intelligent clever people get themselves into trouble by thinking they know more than they do. I'd rather see what the experts say.
Recent decades have seen population estimates forced to be raised by orders of magnitude in many places.