ci(docs): test `cargo doc` warnings and errors (#4635)

* ci(docs): test `cargo doc` warnings and errors

* ci(docs): remove minimal profile

* ci(docs): revert minimal and fix component name

* ci(docs): try cargo doc alone

.github/workflows/lint.yml

@@ -95,7 +95,7 @@ jobs:
         with:
           persist-credentials: false
 
-      - uses: actions-rs/toolchain@v1.0.1
+      - uses: actions-rs/toolchain@v1.0.6
         with:
           toolchain: stable
           components: rustfmt
@@ -108,6 +108,29 @@ jobs:
           command: fmt
           args: --all -- --check
 
+  docs:
+    name: Rust doc
+    timeout-minutes: 30
+    runs-on: ubuntu-latest
+    needs: changed-files
+    if: ${{ needs.changed-files.outputs.rust == 'true' }}
+
+    steps:
+      - uses: actions/checkout@v3.0.2
+        with:
+          persist-credentials: false
+
+      - uses: actions-rs/toolchain@v1.0.6
+        with:
+          toolchain: stable
+          profile: minimal
+          override: true
+
+      - uses: actions-rs/cargo@v1.0.3
+        with:
+          command: doc
+          args: --no-deps --document-private-items --all-features
+
   actionlint:
     runs-on: ubuntu-latest
     continue-on-error: true