Yoiks. This is a really great question...I am sure you knew I would dive right in.
The statement describes the situation pretty well. It's in line with Chomsky's notion that grammar is inherent and innate, and his conversation about self-organizing systems.
Neither diagramming nor parsing are necessary to understand someone's words. But tools such as these serve a different function for me. As you know, my students are described as severely language delayed--generally their eligibility is stated as Autism Spectrum Disorder or Aphasia. So my work is two-fold: it's developmental and remedial, and addresses phonology, morphology, syntax, grammar, semantics, and pragmatics...and voice and fluency and structural anomalies on occasion. For these children, who have almost no structure, and whose language competencies and performance are severely delayed, we use what works--and interestingly, a visual representation of a sentence gives them a handle for even a basic understanding of what a sentence is. In other words, if you were to ask one of my students what a sentence is, he or she would have no clue how to answer that question, and for my severe auts, it's meaningless. But when they begin to put strings of words together in a systematic format, they come to begin to be able to talk about the nature of a sentence...and from there we move into meaning, and shades of meaning, and from there, to written language expression.
I don't know if you have had occasion to see how youngsters in France are doing in terms of written language, but here it is almost inconceivably bad. Written expression is declining not just for children with learning exceptionalities, but in the general population as well. (And of course much of what we read here on Yahoo!A demonstrates that.) It may be that in a couple of decades our oral and written communications will be nothing but truncated words in code--bffs and lols--that is possible, because as you said, language is always emergent.
But one of the first criteria for determining language disorder has to do with intelligibility. As I was writing this response to your question, I recalled a definition about disordered speech that was in the very first textbook for my very first course in language development and remediation. The book was written by Charles Van Riper--known principally for his work with stutterers. He wrote, "Speech is defective when it deviates so far from the speech of other people that it calls attention to itself, interferes with communication, or causes the possessor to be maladjusted." (p.29, and I just noticed that some very naughty dog has chewed the spine of this book!!! Roland, no doubt, because he is still teething...at 3!)
Now, because you are one of the kindest, most accepting, most decent human beings on the planet, and I love you dearly, I am thinking that you may take issue with words such as "defective" or "delayed" or "deviant," not wishing that anyone would be branded that way. Van Riper, also a very gentle soul, and himself a severe stutterer, meant no one any harm with these terms; rather, he was seeking a way to describe language in such a way as to be able to determine when intervention was and was not necessary. We do have to have a way to speak about things comparatively, and sometimes in our efforts to maintain a student's or client's dignity we come up with descriptors that are meaningless...I mean, what the heck is a Specific Learning Disorder or Other Health Impaired to anyone except those who speak edubabble?! But so you don't despair...know that we are always seeking words that do not belittle others. For example, one of the current favorites is "impairment." It will probably fall by the wayside too, eventually.
In any event, there is a role for analysis of grammatical structure; we are not teaching academic language anymore (here), and what is now emerging (linguistically) in many cases meets Van Riper's criteria for non-effective speech/language.
On the other hand, the whole planet seems to be in decline; I suppose I would do better not to "Rage, rage against the dying of the light."
ADDED:
HEMP: I DO hope you will get back to this one. I would love to read your insights from your area of expertise...and not just area of expertise--your understanding as a result of years of experience.