I think it's helpful to view the birth of language as the development technology, with each new word being a new tool. You might imagine that different populations might develop bone knives or wooden spears independently, in slightly different ways; you might also imagine a particularly good design being shared, traded, or spreading to other places. So it probably was with words, and even grammatical strategies: some things radiated from a source at some centre, other things sprung up independently. So all languages are neither sisters nor strangers, but likely something in between.