Bring back documentation publishing to Canary builds

We are using the same approach as in
apache/airflow-site#1055 where we utilise
/mnt directory to store checked out airflow-site.

Fixes: #41931

.github/workflows/static-checks-mypy-docs.yml

@@ -243,9 +243,7 @@ jobs:
       INCLUDE_SUCCESS_OUTPUTS: "${{ inputs.include-success-outputs }}"
       PYTHON_MAJOR_MINOR_VERSION: "${{ inputs.default-python-version }}"
       VERBOSE: "true"
-      # Temporary disabled it until self-hosted ASF runners will be working again
-    if: false
-    # if: inputs.canary-run == 'true' && inputs.branch == 'main'
+    if: inputs.canary-run == 'true' && inputs.branch == 'main'
     steps:
       - name: "Cleanup repo"
         shell: bash
@@ -261,16 +259,23 @@ jobs:
         with:
           name: airflow-docs
           path: './docs/_build'
+      - name: Check disk space available
+        run: df -h
+      - name: Create /mnt/airflow-site directory
+        run: sudo mkdir -p /mnt/airflow-site
       - name: "Clone airflow-site"
         run: >
           git clone https://github.com/apache/airflow-site.git ${GITHUB_WORKSPACE}/airflow-site &&
-          echo "AIRFLOW_SITE_DIRECTORY=${GITHUB_WORKSPACE}/airflow-site" >> "$GITHUB_ENV"
+          echo "AIRFLOW_SITE_DIRECTORY=/mnt/airflow-site/airflow-site" >> "$GITHUB_ENV"
+        working-directory: /mnt/airflow-site
       - name: "Prepare breeze & CI image: ${{ inputs.default-python-version }}:${{ inputs.image-tag }}"
         uses: ./.github/actions/prepare_breeze_and_image
       - name: "Publish docs"
         run: >
           breeze release-management publish-docs --override-versioned --run-in-parallel
           ${{ inputs.docs-list-as-string }}
+      - name: Check disk space available
+        run: df -h
       - name: "Generate back references for providers"
         run: breeze release-management add-back-references all-providers
       - name: "Generate back references for apache-airflow"
@@ -297,3 +302,4 @@ jobs:
       - name: "Upload documentation to AWS S3"
         if: inputs.branch == 'main'
         run: aws s3 sync --delete ./docs/_build s3://apache-airflow-docs
+        working-directory: /mnt/airflow-site/airflow-site