The matter of why Linguists see language as innate is incredibly complex. I don't think I could explain this adequately without derailing the thread. The short version is that we all have a mind that structures language in a certain way. Hence, at an early age we all have an ability to learn any language as we have the ability to structure all languages. The basic idea is that children are able to learn a language without being taught because they have innate knowledge of grammar. They just collect various words from their surroundings, yet because grammar is innate, they know how to put the words in the proper order. In other words, a child only needs to be in the presence of those who speak a language in order to learn a language. He does not need to be taught.
Some people have more talents for learning higher level math than others. Much of the higher level math relies heavily on logic. People who are talented at logic will also be talented at higher level math.
My main idea is this: although children have an innate ability to organize words, or to speak a language by putting words into a proper order, they do not have an innate ability with regard to logical reasoning that parallels this ability. Most people do not learn to use logic well until they are adults and most adults never attain proficiency with this skill. Significant errors of reasoning are common among journalists, businessmen and non-academic writers. Even scholars are known to justifiably accuse one another of committing logical fallacies.
Yes, both grammar and math are left-brained in some loose sense, as they required structured thought. However, the difference is that we have an instinct for grammar, yet we have no such instinct for math or logic. Thus, you do not have to learn the skill of structured thinking to use language in a structured manner because the ability of using a language in this way comes to all humans naturally. Learning to use logic well may help you with this, but you can accomplish this goal just fine without knowing how to use logic well.
The bottom line is that you do not need to have a talent for logic to excel at language, as most novelists and poets are not talented at logic or higher level math.
In recapitulation: If contemporary linguists are correct to claim that much of our knowledge of grammar is innate, then we can conclude that a person does not need to have highly developed logical reasoning skills to provide the proper grammatical structure for his sentences. Thus, if we see a person whose sentences contain impeccable grammatical structure, we should not assume that he is skilled or talented at logical reasoning or any activity that highly emphasizes such reasoning.