His argument does not need beneficial mutations to work. Merely the change in the proportion of gene expression can have profound consequences. Take the breeding of dogs for example. This kind of selection is what he refers to when he talks about bottlenecks, wars, and famine. Small groups with beneficial gene selection could've easily taken over during great changes in societal structure. The idea that evolution can go very fast is pretty well founded in (I forget the name) the idea that you have large stretches of minimal, small improvements, interpuncted by short disruptive events of large improvements/readaptation to a new environment.

Foxes are a better example. Took only 10 generations to domesticate, and the result wasn't just demeanor - physical, dog-like changes resulted as well: https://www.pbs.org/newshour/science/domesticated-foxes-gene...