perf(ci): separate the playwright install and run steps

This allows us to let the servers start while playwright browsers are
downloading, which means we don't need to sleep 50s just waiting for the
servers to start.

Author: joanise

.github/workflows/end-to-end-tests.yml

@@ -99,18 +99,18 @@ jobs:
           npx nx build web-component
           npx nx run-many --targets=serve,serve-fr,serve-es --projects=web-component,studio-web --parallel 6 &
 
+      - name: Install Playwright browsers
+        run: npx playwright install --with-deps chromium
+
       - name: Wait for the servers to be up
         run: |
           # wait for the studio web to be up
-          sleep 50
           curl --retry 20 --retry-delay 10 --retry-all-errors http://localhost:4200 > /dev/null
           # Make sure the back-end API is up too
           curl --retry 20 --retry-delay 1 --retry-all-errors http://localhost:8000/api/v1/langs
 
       - name: Run Playwright tests for studio-web
-        run: |
-          npx playwright install --with-deps chromium
-          npx nx e2e studio-web
+        run: npx nx e2e studio-web
 
       - name: Upload HTML report
         uses: actions/upload-artifact@v6