Make doc CI use nightly (#16147)

# Objective

Fixes #15427
Follow-up to #15428

## Solution

Use nightly and add the environment variables as suggested here:
https://github.com/bevyengine/bevy/pull/15428#pullrequestreview-2331294421

## Testing

CI should run

.github/workflows/ci.yml

@@ -316,7 +316,9 @@ jobs:
             ~/.cargo/git/db/
             target/
           key: ${{ runner.os }}-check-doc-${{ hashFiles('**/Cargo.toml') }}
-      - uses: dtolnay/rust-toolchain@stable
+      - uses: dtolnay/rust-toolchain@master
+        with:
+          toolchain: ${{ env.NIGHTLY_TOOLCHAIN }}
       - name: Install Linux dependencies
         uses: ./.github/actions/install-linux-deps
         with:
@@ -327,7 +329,7 @@ jobs:
         run: cargo run -p ci -- doc
         env:
           CARGO_INCREMENTAL: 0
-          RUSTFLAGS: "-C debuginfo=0"
+          RUSTFLAGS: "-C debuginfo=0 --cfg docsrs_dep"
       # This currently report a lot of false positives
       # Enable it again once it's fixed - https://github.com/bevyengine/bevy/issues/1983
       # - name: Installs cargo-deadlinks

.github/workflows/weekly.yml

@@ -75,7 +75,7 @@ jobs:
         run: cargo run -p ci -- doc
         env:
           CARGO_INCREMENTAL: 0
-          RUSTFLAGS: "-C debuginfo=0"
+          RUSTFLAGS: "-C debuginfo=0 --cfg docsrs_dep"
 
   open-issue:
     name: Warn that weekly CI fails

tools/ci/src/commands/doc_check.rs

@@ -16,6 +16,6 @@ impl Prepare for DocCheckCommand {
             ),
             "Please fix doc warnings in output above.",
         )
-        .with_env_var("RUSTDOCFLAGS", "-D warnings")]
+        .with_env_var("RUSTDOCFLAGS", "-D warnings --cfg=docsrs")]
     }
 }