Generally speaking that's true, e.g, language happens within the brain.
But the processes and mechanisms of language are complicated, and there is no single neurological study which locates all linguistic processes within any specific section of the human brain.
The best that has been posited is that various functions within the brain contribute to the processes of perception, syntax, motor control, and memory, all of which constitute language as a whole.
I can't disagree with anything you've said. The brain is interconnected in such a fashion that "language" as an idea or concept needs more than just the simple areas of the brain dealing with language and speech to correctly communicate in a larger executive context. However, I can kill a specific spot of brain and said person will no longer be able to understand language at all, no matter how well other functions are operating.