[libc++] Add a CI job to build the documentation

At the same time, fix an issue that broke the documentation since 2eadbc86142b.
diff --git a/libcxx/docs/index.rst b/libcxx/docs/index.rst
index 75d5b82..f1c0906 100644
--- a/libcxx/docs/index.rst
+++ b/libcxx/docs/index.rst
@@ -154,7 +154,6 @@
 .. toctree::
    :maxdepth: 1
 
-   DesignDocs/AvailabilityMarkup
    DesignDocs/DebugMode
    DesignDocs/CapturingConfigInfo
    DesignDocs/ABIVersioning
diff --git a/libcxx/utils/ci/buildkite-pipeline.yml b/libcxx/utils/ci/buildkite-pipeline.yml
index 05d6703..faa1081 100644
--- a/libcxx/utils/ci/buildkite-pipeline.yml
+++ b/libcxx/utils/ci/buildkite-pipeline.yml
@@ -202,6 +202,17 @@
         - exit_status: -1  # Agent was lost
           limit: 2
 
+  - label: "Documentation"
+    command: "libcxx/utils/ci/run-buildbot.sh documentation"
+    artifact_paths:
+      - "**/test-results.xml"
+    agents:
+      queue: "libcxx-builders"
+    retry:
+      automatic:
+        - exit_status: -1  # Agent was lost
+          limit: 2
+
   - label: "Legacy standalone build"
     command: "libcxx/utils/ci/run-buildbot.sh legacy-standalone"
     artifact_paths:
diff --git a/libcxx/utils/ci/run-buildbot.sh b/libcxx/utils/ci/run-buildbot.sh
index 1f0b3c7..5502cde 100755
--- a/libcxx/utils/ci/run-buildbot.sh
+++ b/libcxx/utils/ci/run-buildbot.sh
@@ -169,6 +169,13 @@
     generate-cmake
     check-cxx-benchmarks
 ;;
+documentation)
+  export CC=clang
+  export CXX=clang++
+  generate-cmake -DLLVM_ENABLE_SPHINX=ON
+  echo "+++ Generating documentation"
+  ninja -C "${BUILD_DIR}" docs-libcxx-html
+;;
 unified-standalone)
     export CC=clang
     export CXX=clang++