👷 Update build docs, verify README on CI (#10750)

.github/workflows/build-docs.yml

@@ -28,6 +28,8 @@ jobs:
             - docs/**
             - docs_src/**
             - requirements-docs.txt
+            - .github/workflows/build-docs.yml
+            - .github/workflows/deploy-docs.yml
   langs:
     needs:
       - changes
@@ -55,6 +57,8 @@ jobs:
           pip install git+https://${{ secrets.FASTAPI_MKDOCS_MATERIAL_INSIDERS }}@github.com/squidfunk/mkdocs-material-insiders.git
           pip install git+https://${{ secrets.FASTAPI_MKDOCS_MATERIAL_INSIDERS }}@github.com/pawamoy-insiders/griffe-typing-deprecated.git
           pip install git+https://${{ secrets.FASTAPI_MKDOCS_MATERIAL_INSIDERS }}@github.com/pawamoy-insiders/mkdocstrings-python.git
+      - name: Verify README
+        run: python ./scripts/docs.py verify-readme
       - name: Export Language Codes
         id: show-langs
         run: |