The claim of Chomsky's UG is that there is a genetic component to the language faculty of humans. That's pretty much a truism given that we can speak and our closest biological relatives can't. Presumably this cashes out somewhere in our brain.
There's evidence that certain regions of the brain are involved in Language (Broca & Wernicke's areas) given evidence from brain injuries and their effect on language.
He does posit an LAD (language acquisition device) but this is a logical/theoretical construct, to aid in theory building. It seems very likely that we have this given the universality of language in our species and our ability to learn the language of our own culture. The LAD though has not been claimed to be physically located somewhere in our brains by Chomsky (again afaik). Though presumably if we have it, it is fairly likely to be somewhere within our skulls.
This is a non-sequitur because you're implying that such a mutation would be binary: either a creature has it and can communicate with creatures that have the same, or it hasn't and communication is impossible. That's not how evolution works, specialization of organs can happen over thousands of generations. If a specific mutation is beneficial, it spreads through the population over time, and additional mutations in that area can further develop the organ.
Second, our speech apparatus isn't one "organ" in the narrow sense: our lungs/vocal chords/tongue/lips/cheeks/jaw all contribute to the richness of our speech. The breadth of our speaking ability is a specialization of fine muscle control in all those areas, coupled with a finely-tuned hearing apparatus to analyze and understand those sounds.
It doesn't seem like an extraordinary claim to me that our evolutionary path has enhanced our communication ability far beyond our peer species -- because it's clear to see how a rich vocubulary can aid a species' survival over time. I have no knowledge of Chomsky's specific claims, but I don't think you are representing his theory accurately.