Language is a set of behaviors Behavior for humans and other animals requires a brain and body (almost always) The organisation, growth, and function of the brain and body are influenced by genetics.

So we can say first and formost that language is influenced by genetics. Having a tongue, vocal chords, hands, ears. These all help to acquire language.

So now the question is to what degree does genetics influence language and language acquisition?

Is it fully governed by genetics? We know that removing a child from the company of others during development eliminates complex language and severely stunts their ability to acquire language. So, in a single individual, language has never been observed to arise spontaneously and the facility to acquire language is something you can lose.

If you damage certain portions of the brain aspects of language can be removed. See Fluent Aphasia for a particularly odd example. However, these aphasias are known to remit and studies show that other portions of the brain have taken over from the damaged portion. So we know that the capacity for language is not entirely localized to a genetically determined location.

We also know there was a time when there wasn't language and now there is, so at least once in human history language did arise spontaneously. The article you mention is evidence that language can evolve spontaneously in groups of humans. In fact there is evidence that the rudiments of language arise in groups of many animal species including apes, whales, and dolphins. Many animals communicate, but with insufficient sophistication as to be described as a language.

So it is safe to say that there are genetic factors in producing communicative behavior. Sounds, postures, marking etc. I feel it is also safe to say that there are genetic factors that predispose groups of some species to develop more complex systems of communication (but these are not limited to humans) and that groups of humans are particularly good at complex communication.

Unfortunately for Chomsky there is little to no evidence that any one type of language is more likely than another. Grammars, phonemes, words, abstract concepts ... all of these have huge between language variation. The similarities between languages are well explained by either physical characteristics of speech production or regularities in the environment of the languages origin.

So yes, language is both learned, and cultural, even if it isn't purely so.