Did all human language originate at the same place (in southern Africa)?

New Zealand biologist Quentin D. Atkinson has an article that's getting a phenomenal — phoneme-able? — amount of attention. The NYT says:

[Atkihnson] has found a simple but striking pattern in some 500 languages spoken throughout the world: A language area uses fewer phonemes the farther that early humans had to travel from Africa to reach it. Some of the click-using languages of Africa have more than 100 phonemes, whereas Hawaiian, toward the far end of the human migration route out of Africa, has only 13. English has about 45 phonemes.

This pattern of decreasing diversity with distance, similar to the well-established decrease in genetic diversity with distance from Africa, implies that the origin of modern human language is in the region of southwestern Africa...

Paleoanthropologist John Hawks asks:

Why should the origin of languages have had the largest inventory of phonemes? If small populations typically lose phonemic variation, why would sparse hunter-gatherer populations of Africa have built up the largest store of sounds just as they were getting started talking?

Perhaps when people were first talking, they made a lot of sounds, more like animal sounds. Then came the breakthrough: words. Once the idea of words arrived, you didn't need so many sounds. You can get lots of words out of very few sounds, differently arranged. The attention shifted to word making, and people didn't struggle to come up with more sounds. Indeed, they took advantage of the easy ones everyone could say and shed the extras.