Judging by ontogenetic development, language is derived from two separate streams (and a third one later on).
One of these is the development of object and action concepts by visuomotoric handling as opposed to background, space or situation. This may well be specific to humans with infants developing eye-hand-coordination in contrast to other species which do not undergo such a phase and may not develop strong concepts for objects. Infants also form a concept for actions when they experience their own agency and extend it to other agents.
The other is the sound-making ability, which appears innately pleasurable to human infants, and contributes to a long period of babbling.
At around 10-12 months of age these developments combine (‘naming insight’ in child language literature) and articulated words which refer to simple concepts arise. These refer to objects at first, sometimes actions and are articulated by phonological sequences. The one and two word (‘pivot grammar’) stages follow.
I see no reason to assume that phylogenetic development should have been different – the development of articulatory abilities, considered beautiful for their own sake (‘music/song’) running in parallel with solid conceptual structuring of the environment before one and two word communication became commonplace.
What about communication and communicative needs? I believe they popped up after the first concept naming skills took root. Suddenly there must have been a drift towards higher information content messages rather than just a string of words denoting objects and actions. Communication then is what must have driven grammar. Note that grammar remains a highly social accomplishment – like phonological sophistication it even requires a critical period. Grammar like phonology probably has a strong striatal component – habit learning – and it spontaneously appears in sign language as well. For this we would require a more detailed theory on how grammar arose from communication, when each grammatical system is distinct from any other.