chore(ci): Use cargo-docs-rs to make docs check with the same options that would be used by docs.rs

Author: tottoto

.github/workflows/CI.yml

@@ -308,9 +308,18 @@ jobs:
     - uses: Swatinem/rust-cache@v2
     # we exlude crates that do not use rust-version = { workspace = true }
     - run: cargo no-dev-deps --no-private check --all-features --workspace --exclude grpc --exclude tonic-protobuf\*
-    - run: cargo no-dev-deps --no-private doc --no-deps --all-features --workspace --exclude grpc --exclude tonic-protobuf\*
-      env:
-        RUSTDOCFLAGS: "-D warnings"
+
+  doc:
+    runs-on: ubuntu-latest
+    steps:
+      - uses: actions/checkout@v5
+      - uses: dtolnay/rust-toolchain@nightly
+      - uses: dtolnay/install@cargo-docs-rs
+      - uses: taiki-e/install-action@cargo-hack
+      - uses: Swatinem/rust-cache@v2
+      - run: cargo +nightly hack --no-private docs-rs
+        env:
+          RUSTDOCFLAGS: "-D warnings"
 
   test:
     runs-on: ${{ matrix.os }}