We are a commune of inquiring, skeptical, politically centrist, capitalist, anglophile, traditionalist New England Yankee humans, humanoids, and animals with many interests beyond and above politics. Each of us has had a high-school education (or GED), but all had ADD so didn't pay attention very well, especially the dogs. Each one of us does "try my best to be just like I am," and none of us enjoys working for others, including for Maggie, from whom we receive neither a nickel nor a dime. Freedom from nags, cranks, government, do-gooders, control-freaks and idiots is all that we ask for.
Diagramming sentences, in itself, is not particularly useful. However, the knowledge that allows one to do the task is essential. In other words, diagramming sentences is really just a pedagogical tool to demonstrate the knowledge.
My son is in college and has rented a 2 bedroom apartment. He has been seeking a roommate. His ads are very straight forward: "3rd year student in X seeks roommate for 2 BR, 1 BA apartment". Some additional details follow.
He has been stunned by how many respondents are unable to understand very basic sentences. Guys that want to bring their wife & 2 kids, folks that think they are going to get the entire apartment (for 1/2 the market rate).
I would argue that people are getting stupider because they are not taught how to communicate via written English.
Writing has value only as a tool for communication, and communication for specific purposes. Your son's advert is similar to the "telegraphese" of an earlier day (price per word), or how modern radio operators use a common shorthand (Q codes or abbreviations like XYL or 73). Regardless, the human mind understands words and grammar even when mangled -- you'd have to ask a linguist why.
What is the point of knowing how to diagram a nongrammatical sentence whose meaning is perfectly clear? e.g., "batter up!" or diagramming powerful rhetorical devices that defy grammar such as epizeuxis, e.g., "Reputation, reputation, reputation!" Othello, 2, 3.
On every associate's desk in my firm is a copy of Strunk & White's Elements of Style. So they are at least competent writers in modern business and legal English. As an aside, words like "hereto" are banned. I doubt any of them could diagram a sentence, and they all communicate perfectly well in the shorthand text on their smartphones.
The proof of competency in writing is did the judge grant your motion, did the customer sign the contract, did your story get published. It is not whether one can diagram a sentence.
If you want to polish writing beyond the competent stage, get a copy of Farnsworth's Classical English Rhetoric.
I write tons of technical reports. Every day, I think God for the nuns in third and fourth grade who beat grammar, spelling and sentence structure into me. Diagramming sentences was a huge part of that and has helped me tremendously over the years. I write a proper sentence now without even thinking about it.
I also read almost as many reports as I write. I am appalled at supposedly educated people, engineers and lawyers, whose command of the written English language is tenuous at best.
I actually had one young engineer submit a report was full of text speak. He no longer works here.
At some point, diagramming sentences was not taught any longer. My guess is it was sometime in the 70s. I noticed a distinct difference in the writing abilities of older people compared to younger people.
I was not taught sentence diagramming, and also felt that it was very cumbersome. When I was homeschooling my daughter, however, I found a rival system for analyzing sentences and teaching grammatical constructions in a program available for free on the Internet: KISS Grammar (developed by Ed Vavra). I won't provide a direct link, but you can easily search on it. Basically the student marks up the sentences with underlines, circles, boxes, etc. The student is able to analyze much more complicated sentences (the text uses sentence from real literature) that is feasible with diagramming. Over a period of several years in grade school, my daughter learned far more grammar than I ever did through high school!