workflows: linter does not lint docstrings

The linters and pre-commit configs do not run the `cargo doc` and fail on docstring formatting issues like
```
warning: unclosed HTML tag `String`
--> cli/src/status.rs:43:10
|
43 | /// A Vec where each element is an overlay name from the input ...
| ^^^^^^^^
|
```

Inspect the repository's workflow, linter, and pre-commit configuration, then run the existing checks and `cargo doc` to reproduce the warning shown for `cli/src/status.rs`. Update the relevant configuration so documentation warnings such as unclosed HTML tags are checked automatically, and confirm the checks fail on the example issue.

Written by the indexing model from the issue text.