Run the docs site's broken link checker in the docs linting CI job

- We had `rake test` as a task, but we never actually ran it in CI or
  anywhere before building the site.
- The `html-proofer` gem
  [only supports Ruby 3.1](https://github.com/gjtorikian/html-proofer/blob/main/html-proofer.gemspec#L20),
  so I bumped the linting job to use Ruby 3.1 across the board.
- This will make things slower (or maybe it's taking ages because of my
  dodgy hotel internet connection), maybe we should only run it on a
  schedule as a separate job?
- Fixes https://github.com/Homebrew/brew/issues/ 15908.