Certain writers do. It just depends on the domain. For book writers, there are tools like Scrivener. For screenwriters, there are tools like Final Draft. These tools help tremendously with auto-formatting and version control. They also keep track of things like characters and settings, presumably by assigning them to certain classes and IDs "under the hood." (I've never looked under the hood, but I assume they are XML-based.) They are fantastic tools for the writers they're made for. They're also fairly WSIWYG, and do not require writers to know anything about markup languages. That's pretty key, because many writers are not technically proficient. (Modest technical proficiency, at the very least with HTML and some CSS, is probably going to become a requirement in the future...but I digress.)
For journalism, however, Word is the default—and obviously, it has its strengths and its weaknesses. Wonky changelogs and substandard version control are two of the biggest. Some editors at some publications are moving over to Google docs, for the superior versioning and collaboration capabilities. But Google's editor doesn't offer the robustness and feature set of Word, and it's unfamiliar, so a lot of editors and writers are reluctant to adopt it. Personally speaking, I much prefer Google to Word when it comes to working with my editors. I'll gladly take in-doc comments and version control over emailing Word docs back and forth, any day of the week.
Basically, they don't know there's a better way. They're used to hacking through Word's track changes, and they always have a deadline to meet. That doesn't leave much time for improving your toolset.
Also, many writers have very idiosyncratic workflows. From what I gather their education and workplaces focus more on the craft of writing well, regardless of whether it's with a pencil or a keyboard. Whereas it's hard to contribute to a codebase without being exposed to version control.
(I vaguely recall him mentioning that on his app even)