It's well established humans evolved in Africa and migrated out many times. Hominids have been in Africa and Asia and Europe for millions of years, and modern humans mixed with them some, giving today's regional varieties.
It's also fairly clear that modern human behavior was there in force 40,000 years ago and didn't seem to be there at all 80,000 years ago. And that behavior is cultural, passed along by speech. It's as if people didn't have language 80,000 years ago but they all did 40,000 years ago. They reached Australia about 60,000 years ago, complete with language and cave painting, limiting it more to 60,000 to 80,000 years ago.
Waves of modern humans left Africa around 70,000 to 50,000 years ago and overran everything else. All the existing hominids bred with modern humans and/or died off. Everywhere out of Africa, these modern humans dominate. So it's natural to assume the ability to speak, or at least speak easily, started in one spot (around Ethiopia) around 70,000 years ago and spread out from there.
But in Africa there are still humans that branched off 100,000 or 200,000 or 300,000 years ago. For example the Pygmy and the San. And they can speak as easily as anyone else. The Bushman clicks, you might argue they can speak better than everyone else. They can make art too. Though, traditional Pygmy art is curiously stick-figurish.
Jared Diamond is a proponent of the theory that language started about 40,000 years ago, and is responsible for human culture. He pointed out that Creoles (languages spontaneously created by children without a common language) have certain grammatical structures in common, for example not not still means not. Noam Chomsky pointed out that actual languages use only a particular subset of all theoretical grammars. So there's some vague default language structure in our brains.
So here's my theory:
We don't know what these genes are yet. We've been tracking FOXP2, but that is shared with Neanderthals, so that's not it. But I predict that there ARE language genes, they were strongly selected for between 80,000 years ago and now, and we will find them. There probably aren't many of them, one or two or maybe a dozen. Outside of Africa, we got modern humans mostly replacing the Neaderthals and Denisovans and Homo Erectus etc with just a little mixing. Inside Africa there was a lot more mixing. You only need the necessary few language genes to get spread, the rest can be whatever, so you don't need to replace populations for these genes to still spread. That's how the San could have diverged over 100,000 years ago but still have obtained the genes to speak easily that only evolved 70,000 years ago.
There is also selective pressure for those previous hominids who could learn to speak with effort to get better at it. So you may find some alternative mechanisms for speech in the San and Pygmy and Africa in general. Maybe allowing them some forms of speech the rest of modern humans can't easily learn.
The place to hunt for these genes is in the populations of Africa. Outside Africa, all modern humans inherited most of their genes from the group that could speak, so you can't tell which ones are specifically speaking related. Inside Africa, in some populations most genes are NOT from that group, so the genes that ARE from that group are the ones to look at.
Later revisions: