Well, I'd be the last person to argue that one cannot write unless one can diagram. That said, though, I think that diagramming can be an extremely useful tool, especially for visual learners (I readily confess that I am one!)
I agree that it is critical for writing success to write a lot and read a lot, but there can be an important place for really understanding our language and how the parts work together. This knowledge can be gained in other ways, for example intensive foreign language study. But even there, I see a real value in diagramming. If you want to take a complicated sentence and put it into Latin, it can be really useful to be able to visually mark out what goes where--how are the parts related? What can I put in nominative case, what will need objective, or genitive, or ablative . . . and I can personally "see" and identify this much more quickly if I sketch out a quick diagram first. Also, if I am writing and having trouble with a sentence, one of the first tools I turn to to diagnose my problem is a diagram. I can often see at a glance why I am having a difficulty, and can then correct that.
But I really miss something in a lot of current writing today. There is just something missing in many of today's writers, when one compares writers with C.S. Lewis, or Auden, or Orwell . . . . IMO slapdash English just doesn't come across as well as disciplined English. Perhaps it's just an aesthetic bias I have, but there is just a class of writing that is set apart . . . and without fail, one finds that THOSE types of writers knew their stuff--they were well read, they wrote a lot, but they also parsed and diagrammed and translated Latin poetry and tried to write their own . . . and I think it shows. Now, I'm not sure I'm raising any professional writers, but I still thought it was a tool worth learning. And my children, none of whom can be accused of an irrational fondness for grammar, LOVED diagramming. It is not difficult to learn and has a real intellectual challenge--like a puzzle. We diagram things for fun sometimes at dinner. OK, maybe we're just weird. (Maybe???)
One other note--when I was in graduate school and grading college freshman papers, I decided to run my own little (decidedly unscientific) experiment. I held aside those papers and exams I got that were really special--super writing, writing that just stood above the rest. I made it a point to ask all those kids a few questions. And the results? Almost without exception, all those kids had seriously studied a foreign language (at least 3+ years of the same one), all of them had had to write essays often in high school (like several a WEEK) AND all of them had had significant grammar instruction, often including diagramming. Just something I found very interesting and that I have tried to keep in mind in preparing my own kids for college. Can you become a very decent writer without diagramming? Absolutely. Does it still make sense to dedicate a small space of time to learning as useful a tool as diagramming? Again, IMO, absolutely.