"For the out of Africa theory to hold water, the first population would have to have been very small."
Would the original population have to have been small, or could the same effect been caused by a "bottleneck" further along the line. The great Toba volcanic event 74,000 years ago is thought by some scientists to have reduced human population to no more than 5 or 10 thousand individuals. The caldera left by Toba measures something like 18 miles by 65 miles. By way of comparison Pinatubo left a crater 3 miles in diameter. Big, bad "nuclear winter".
I think the sentence refers to the Replacement model, which sez that there was only one population of modern humans, it originated in Africa, and spread out across the Earth, overland, extinctifying all other species of homo (and a great many non-human species besides), beginning about 50,000 years ago, or less.
The alleged bottleneck was 24,000 years or so earlier than that.
The model doesn't hold water regardless.