The more serious formulation of the hypothesis is that communication is superimposed on a “language” capacity that evolved for cognition.

To take a completly non controversial example of how that could work, consider the more mechanical side of how language works. Our respiratory system evolved to exchange co2 and o2 with the atmosphere. Our mouth evolved to be the first stage of our digestive system. Our ears evolved for general sensory input. None of those systems evolved for communication, but are central to the mechanics of how we communicate.

In the same way, the idea was that we developed a set of genetal purpose cognitive tools, and the only thing specific to communication was the ability to serialize de-serialize those thoughts into soundwaves.