Why? It doesn't seem to me that their theory hinges on spear-throwing did in all the mega-fauna. The Neanderthals could have been outcompeted without that happening.

Paul Bingham of Stony Brook has a theory about how throwing was, for another reason, a big step in human evolution:

The problem was answered when the cost of coercing a cheating individual in a cooperative effort, otherwise known as the free-rider problem, was lowered. This happened when a species developed a way to threaten adult con-specifics from a distance, specifically by evolving the ability to throw, thus exploiting the diffusion of risk intrinsic of actors utilizing Lanchester's Square Law. He proposes that we evolved the ability to repel predators and scavenge their kills in the African savannah. It was later adapted as threat projection towards free-riding con-specifics in non kin cooperative groups.