I should add that one of the issues with language acquisition is that it appears to happen during a fairly narrow window, that acquiring languages has fairly significant neurological effects, but that if it isn’t acquired fairly early on (probably around the age of five), a person can never really acquire full human language (the recursiveness you mention, as well as displacement, arbitrariness and semantics). So language isn’t just a matter of genetics or of initial neural structure (as found in utero or at birth), but also is heavily affected by post-natal development.
One thing that linguistics have observed is that creoles evolve and develop mainly in children, that while adults of different linguistic heritages can cobble together functional pidgins, those pidgins are something akin to “half languages”, whereas a creole has a far more consistent and regular grammar and other linguistic structures, and while based on features of parent languages (sometimes from completely different language families), they also acquire very novel features not found in parent languages. The general view is that children, with their capacity to learn language, also have significant abilities to modify and enlarge languages. The older we get, the less agile and adaptable our linguistic abilities become. It’s why a child who learns a second language in a very immersive setting can often acquire that second language to a level of a near-native speaker, but if I were to learn, say, Chinese now in my late-40s, I’d probably do well enough at simple exposition and requests, I’d never be able to speak it at the level as when I learned it in childhood.
The long and the short is that while there’s a clear genetic component to language, the best genes ever give any trait is a bit of a launching pad, but how the rocket flies is greatly dependent upon the environment in which we develop. It’s conceivable that the raw neurological machinery of language evolved at some time prior to actual human language evolving, and that actually putting the genetic, neurological and developmental pieces together took more time. We do know from the archaeological record that fully modern humans (at least in appearance) didn’t behave that much differently than their recent cousins and ancestors, and we don’t really see the first glimmers of what we would view as human culture (ie. symbolism, ritual) until tens of thousands of years after H. sapiens first appears on the scene. For a long time our behaviors and toolkits could hardly be distinguished from, say, Neanderthals. My guess is the difference is language, which facilitates cultural transmission with far greater accuracy than primate communication systems seen in other hominoids.
