The type of innate grammar you're describing is known as "universal grammar," and was one of Chomsky's seminal contribution to linguistics. He modeled all gramm

The type of innate grammar you're describing is known as "universal grammar," and was one of Chomsky's seminal contribution to linguistics. He modeled all grammars as ~88 binary parameters such as head marking vs. tail marking, ergative-absolutive or nominal-accusative alignment, double negation etc. It was very successful, though cracks are appearing like the controversial absence of relative clauses in Pirahã, which are supposed to be universal (under X-bar theory, which is related to the above.) I have my doubts; I still cling to Whorfian ideas of linguistic relativity and suspect the governance and binding framework doesn't explain more nuanced or culture-specific aspects of syntax like comparisons or kinship systems, but it's hard for me to argue with the success they've had in schematizing a ton of language families under a simple-ish model. Disclosure: not a linguist, but I play at one.