I was thinking about my earlier “story-telling ape” comment, and wondered, what if it started with some form of crude mime to communicate plans for collective hunts, and bragging about successful hunts afterwards – the latter also serving the purpose of preserving knowledge about how to hunt successfully. The mime could easily involve initially-random excited vocalisations, with these vocalisations becoming ritualised over time, and starting to form part of the mimed communication, in turn forming the crude basis for a language. This improved communication and hunt coordination could easily have provided a selective advantage for those with a tendency and/or aptitude for such communication, and natural selection could have gradually snowballed this into language-use.
I’m not an anthropologist, let alone an evolutionary anthropologist, but this ‘just so’ story does seem to offer a possible explanation for how the whole language thing got started.
I don’t think the “crude mime” idea would be outside the realms of possibility for a cousin species to the chimpanzee.