docs(ci): Add versioning support without using 'mike'

It seems it may have be simpler to just use 'mike'..

Additionally squashes related commits providing minor fixes + improvements:

- Use a job dependency (`needs`) to avoid `push` event race conditions due to parallel jobs.
- Improve workflow file documentation via inline comments.
- Make ShellCheck linting happy.
- `chown` doesn't seem to work unless on the default branch for CI. Opted to use the docker `--user` approach instead.

docs/mkdocs.yml

@@ -17,6 +17,13 @@ theme:
     - navigation.tabs
     - navigation.expand
 
+# We do not use `mike`, but enabling this will enable the version selector UI.
+# It references `versions.json` on `gh-pages` branch,
+# however we have a basic setup that only matches `version` to a subdirectory.
+extra:
+  version:
+    provider: mike
+
 markdown_extensions:
   - toc:
       permalink: ⚓︎