Here's a paper that makes this argument: http://www.ling.upenn.edu/~ycharles/PNAS-2013-final.pdf
I'll take a look at the paper you linked, but I'm also curious, how does your technique make a distinction between rule-based grammar being fully learned vs mostly or partially learned? That is, how can you say that improved grammar over time is evidence against innate ability to understand/use grammar? It seems we have an innate ability to walk in that we seem to be "wired for it", but we are still terrible at it for a bit over a year on average. The fact that we get noticeably better over time doesn't seem to rule out innate ability. This of course assumes "innate ability" doesn't mean "completely unlearned".
Given this reframing of the research, I fail to see how this study demonstrates anything, though. If the brain has special structures that allow humans to learn grammar, you would still expect to see the same pattern of "terrible, broken grammar -> mediocre grammar -> good grammar" that we see. We see the same learning pattern for things that the brain does have special structures for, such as... well... everything from sight to motor control.
There are a number of considerations at play here, and a long history of violently disagreeing about them.
Is language tied to grammar? Or is grammar tied to language[1]? There has been a lot of work seeking to examine the common underpinnings of languages that seem otherwise disparate, and a somewhat flexible core grammatical structure has been proposed as one of these unifying components. The speculative step some take next (including people like Chomsky) is that the reason that all of these common patterns appear is that the brain has evolved some specific hardware that serves as the basis for a set of communication protocols that we call natural languages, and all natural languages instantiate this basic hardware in different ways[2].
If you think this kind of hardwiring is possible, then another obvious-but-not-obvious question follows. Can language really be learned[3]? Can you take a clean slate of neurons (some arbitrary configuration of connections) and, given enough input data, end up with a system that understands language (to frame it as a machine learning problem, which it may or may not be)? Or is it closer to a game of connect-the-dots; the bare framework is there, built into the way the brain arranges itself over the course of millions of years. Input data is just required to figure out exactly what the final arrangement of the picture is, whether it looks like English or Russian or Mandarin.
The current mood in cognitive neuroscience academic circles seems to be the former case, that you can 'simply' throw enough data at a general-purpose learning model and get a language-speaker out. Partly due to the success to deep learning approaches in what were previously very difficult problems in NLP. But I don't think it's fair to say there's nothing worthy of debate here.
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[1] Or is this a false dichotomy and there's some even weirder relationship between the two.
[2] For a primitive example, languages with SVO ordering vs. SOV ordering. Different instantiations of the same triple with subjects, objects, and verbs. Ray Jackendoff took this another step with formulations like XBar grammar, which was trying to demonstrate how a universal, parameterized grammar might look.
[3] Grice had a fun theory that the observable signal of language is _so_ noisy that it's impossible to learn outright, regardless of the number of examples you're exposed to. And thus, everyone speaks a slightly different language, and some sets of these idiolects are, roughly, mutually compatible. I don't think he ever did any math to demonstrate this though.
I'm not sure how you'd reduce this to something mathematically demonstrably, but it seems to me that it's the only thing that could be true. The alternatives is that languages are real, distinct, concrete things that exist independent of people, rather than arbitrary lines drawn by people around the continuous and evolving variation in ways people communicate.
Of course, there are plenty of assumptions in that model (for one, can you even separate 'language' from 'sentience' like an information model would suggest. I'm not so sure you can, but then I've always been somewhat sympathetic to Sapir-Whorf). I'm also not a competent enough mathematician in these fields to see where all the holes in this line of thinking are.
Dogs can learn. They just can't learn grammar. Why? Because the human brain is primed with grammatical primitives that a dog brain hasn't got.
Personally, I think that's a bunk explanation, but this conclusion is not obvious either way.