No. Innate grammar has always been about how humans aquire language, not how any possible system which understands human language must posses that innate grammar.
Therefor a human with no understanding of grammar/language, and using no innate biological circuits, could process grammar and respond with language.
The flaw in this argument would be how to teach a human to do this without grammar ...
> We may as well do it the way LLMs do
We almost certainly don't learn the way LLMs do, it's just too data inefficient.
And I don't see what current LLMs can say about a universal grammar in the Human brain, unless there is proof that a LLM-style attention mechanism exists in the brain, and that it is somehow related to language understanding.
LLMs on the other hand can easily learn these non-human grammatical structures which means that they are not the way humans do it.