[ci] Cache playwright browsers (facebook#32678)

No reason to download them from scratch every time.
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[//]: # (BEGIN SAPLING FOOTER)
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with [ReviewStack](https://reviewstack.dev/facebook/react/pull/32678).
* facebook#32680
* facebook#32679
* __->__ facebook#32678

Author: poteto

.github/workflows/runtime_build_and_test.yml

@@ -566,6 +566,24 @@ jobs:
         run: rm -rf build
       - run: yarn install --frozen-lockfile
         if: steps.node_modules.outputs.cache-hit != 'true'
+      - run: yarn --cwd fixtures/flight install --frozen-lockfile
+        if: steps.node_modules.outputs.cache-hit != 'true'
+      - name: Store Playwright version
+        id: playwright_version
+        run: |
+          PLAYWRIGHT_VERSION=$(npm ls @playwright/test | grep @playwright | sed 's/.*@//')
+          echo "Playwright's Version: $PLAYWRIGHT_VERSION"
+          echo "playwright_version=$PLAYWRIGHT_VERSION" >> "$GITHUB_OUTPUT"
+      - name: Cache Playwright Browsers for Playwright's Version
+        id: cache_playwright_browsers
+        uses: actions/cache@v4
+        with:
+          path: ~/.cache/ms-playwright
+          key: playwright-browsers-v6-${{ runner.arch }}-${{ runner.os }}-${{ steps.playwright_version.outputs.playwright_version }}
+      - name: Playwright install deps
+        if: steps.cache_playwright_browsers.outputs.cache-hit != 'true'
+        working-directory: fixtures/flight
+        run: npx playwright install --with-deps chromium
       - name: Restore archived build
         uses: actions/download-artifact@v4
         with:
@@ -574,16 +592,6 @@ jobs:
           merge-multiple: true
       - name: Display structure of build
         run: ls -R build
-      - name: Install fixture dependencies
-        working-directory: fixtures/flight
-        run: |
-          yarn install --frozen-lockfile --cache-folder ~/.cache/yarn
-          if [ $? -ne 0 ]; then
-            yarn install --frozen-lockfile --cache-folder ~/.cache/yarn
-          fi
-      - name: Playwright install deps
-        working-directory: fixtures/flight
-        run: npx playwright install --with-deps chromium
       - name: Run tests
         working-directory: fixtures/flight
         run: yarn test