Drop Docker and remote ccache usage from publish_website.yml. (#18421)

Progress on #15332 - one less
workflow needing the `gcr.io/iree-oss/base` dockerfile and a ccache
storage on GCP.

Tested on my fork:
* Cold cache (5m30s):
https://github.com/ScottTodd/iree/actions/runs/10690774251/job/29635889592
* Warm cache (3m30s):
https://github.com/ScottTodd/iree/actions/runs/10690871405/job/29636198158

skip-ci: no impact on other workflows

.github/workflows/publish_website.yml

@@ -44,30 +44,25 @@ jobs:
         with:
           python-version: 3.x
           cache: "pip"
-      - id: "gcp-auth"
-        name: "Authenticating to Google Cloud"
-        uses: "google-github-actions/auth@v1"
-        with:
-          token_format: "access_token"
-          credentials_json: "${{ secrets.IREE_OSS_GITHUB_RUNNER_BASIC_TRUST_SERVICE_ACCOUNT_KEY }}"
-          create_credentials_file: false
       - name: Installing dependencies
         run: |
           pip install -r docs/website/requirements.txt
           pip install requests
+          sudo apt update
+          sudo apt install -y ninja-build
+          echo "CC=clang" >> $GITHUB_ENV
+          echo "CXX=clang++" >> $GITHUB_ENV
       - name: Generating release index
         run: |
           ./build_tools/scripts/generate_release_index.py \
             --repo="${GITHUB_REPOSITORY}" \
             --output=docs/website/docs/pip-release-links.html
+      - name: ccache
+        uses: hendrikmuhs/ccache-action@v1.2
+        with:
+          key: ${{ github.job }}
       - name: Building documentation files
-        run: |
-          ./build_tools/github_actions/docker_run.sh \
-            --env "IREE_CCACHE_GCP_TOKEN=${{ steps.gcp-auth.outputs.access_token }}" \
-            --env "IREE_WRITE_REMOTE_CCACHE=1" \
-            --env "CCACHE_NAMESPACE=gcr.io/iree-oss/base@sha256:dc314b4fe30fc1315742512891357bffed4d1b62ffcb46258b1e0761c737b446" \
-            gcr.io/iree-oss/base@sha256:dc314b4fe30fc1315742512891357bffed4d1b62ffcb46258b1e0761c737b446 \
-            ./docs/website/generate_extra_files.sh
+        run: ./docs/website/generate_extra_files.sh
       - name: Setting git config
         run: |
           git config --local user.email "iree-github-actions-bot@google.com"