Pythagorean said:
To be fair, Vygotsky would not require grammatical structure to be a cultural habit rather than genetically hardwired. His position only requires that the habits of thought which create the higher human functions are encoded by language and evolved socioculturally.
The real problem with a hard Chomskian position is that because it requires genetic hardwiring, this does not fit with either the available paleo or neuro evidence. Langauge arose too fast in human history, and there is too little evidence of a syntax template in the brain's architecture, and so a more soft-wired position seems necessary.
What is actually universal in languages is not word order but an idea - the idea of a sentence that involves the three components of a subject/verb/object, a linear causal tale of who did what to whom. This is what ties every language together. Subject, verb and object can come in any order, but these three components act as a mental template that breaks reality into crisp causal statements.
Yet it is in turn still unknown how the brain could be hard-wired, or even just "language-ready", to see the world in terms of subject/verb/object relationships. Again, it makes no real difference to the Vygotskean story whether this universal structure is hard, soft or un-wired (and completely learned and handed down as a cultural habit). But it is an important research question in its own right.
As an aside, given all the Heidegger talk, it is worth noting that the inauthentic view seems in fact fundamental to humans in this regard. Langauge clearly objectifies the subject, the doer, along with the doings and the done-to (the verbs and the objects). It already lifts us out of any local particular notion of the subject, the active agent, the effective cause, and forces us into a generic or objective stance where we are just an example of such a locus of agency, the cause that produces the effects.
The open question is whether animals also have some kind of proto-objectification and cause and effect thinking wired in. I would expect this to be so as the neural architecture of apes is so similar.
This is then why you need an explanatory mechanism such as Pattee's epistemic cut to explain how brains that are basically the same can with just a little tweaking start to operate at a whole new level.
In animals, syntax and semantics would all be jumbled together in unstructured fashion. In humans, they have become crisply divided into "a universal language generator" (even if the template is more cultural than neurological) and a vocabulary (syllables coding for ideas and impressions).
. They have an interest in making papers seem ground-breaking when they aren't.