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.

.github/workflows/docs.yml

@@ -32,7 +32,7 @@ jobs:
       - name: Install Ruby
         uses: ruby/setup-ruby@v1
         with:
-          ruby-version: "2.7"
+          ruby-version: "3.1"
           bundler-cache: true
           working-directory: ${{ steps.set-up-homebrew.outputs.repository-path }}/docs
 
@@ -40,9 +40,11 @@ jobs:
         working-directory: ${{ steps.set-up-homebrew.outputs.repository-path }}/docs
         run: bundle exec rake lint
 
-      - name: Build docs site
+      - name: Check for broken links, then build docs site
         working-directory: ${{ steps.set-up-homebrew.outputs.repository-path }}/docs
-        run: bundle exec rake build
+        run: |
+          bundle exec rake test
+          bundle exec rake build
 
   rubydoc:
     if: github.repository == 'Homebrew/brew'