That's what the scare quotes around "unnatural" are for. But it's hotly debated whether grammar does indeed originate in "years of exposure to other speakers of language". Certainly some aspects of grammar are language-specific, but many appear to be universal, or close to it. From the paper I linked above:
For example, adjectives that denote quality have been argued to precede adjectives conveying size, which
in turn precede adjectives conveying shape, and so on, in all languages (5). Similar claims have been
made for other adjective types, and the respective ordering restrictions are given in (6).
This is one of the founding principles of today's dominant Chomskyan linguistics, perhaps best argued popularly in Stephen Pinker's The Language Instinct.
Not sure Chomskyan is dominant, but anyway, yes, an ability to be grammatical, is widely regarded as innate. Sort of like how every computer can compute the results of a Turing machine, regardless of the actual machine language or high level programming language glommed on top.
Or perhaps to say that humans parse a sentence like a concatenative/stack language, but not an algol/C-like language. (Not saying exactly that, but saying that's the sort of statement that could get to the heart of it).
Whereas "C vs Java" varies across cultures".
That is, the low level hardware memory model is universal, but the shape of the structures on top are diverse.
I would also highlight a distinction between universal grammatical rules that are key to "making any sort of sense at all" vs customizable rules that "disambiguate among multiple possible interpretations".
That was put poorly, but bear with me.
That is, SVO vs OVS is arbitrary, and could go either way, but members of each community need to agree. But number-color-object is nearly universal because color-number-object is uninterpretable as a concatenation of color with number object, unless a language specifically develops a concept of a color applying to more than one non-group entity. And here I speculate, but I can easily imagine a human language where an adjective before a number is understood to emphasize that all items are exactly the same color, or that the items are all connected with no other-color between them, like a "red 3 plates" understood to mean the plates are in a stack.
Heck, I bet that if I were really good looking or famous and started using that pattern, it would spread
Chomskyan theory is less dominant than it used to be. Before WWII, linguistics was largely the study of European languages, and as a result, there seemed to be a lot of similarity between languages. Chomsky was on the tail end of this.
In recent years, the amount of research of non-European languages has exploded and we've realized that languages are much more diverse than we thought. For example, it's not just a matter of SVO, OVS, VOS, or any permutation of the previous; some languages don't even use the subject-object pattern(they do something called ergativity).
TLDR: there may be some sort of innate grammar, but it is much, much more murky than we used to think.
For example, adjectives that denote quality have been argued to precede adjectives conveying size, which in turn precede adjectives conveying shape, and so on, in all languages (5). Similar claims have been made for other adjective types, and the respective ordering restrictions are given in (6).
This is one of the founding principles of today's dominant Chomskyan linguistics, perhaps best argued popularly in Stephen Pinker's The Language Instinct.