ci: cache Playwright browsers keyed on @playwright/test version (#489)

The Playwright browser download (Chrome for Testing + Chrome Headless
Shell + FFmpeg, ~170MB total) is the single largest cost in the
start-app composite action. Observed in a recent E2E run: 4m15s to
download cold vs ~10s on a warm cache. Across weekly cache evictions
and the ~30% of runs that hit a cold cache, that is multiple action
minutes per PR averaged over time, and several minutes of wall-time
regression on any individual cold run.

Split the existing `playwright install --with-deps chromium` step
into two halves:

1. `playwright install-deps chromium` runs unconditionally because
   apt packages live under /var and are not preserved by actions/cache.
   This is fast (~10s).

2. `playwright install chromium` runs only when the browser cache
   under ~/.cache/ms-playwright misses. The cache key is pinned to
   the @playwright/test version read from the root package.json, so
   a future bump invalidates the cache automatically and the old
   browsers are replaced cleanly.

Sharding (the other half of the original optimization plan) was
evaluated and rejected. With ~80s of per-job setup and only ~60s of
test time, 2-way sharding would roughly double action minutes while
saving only ~5s of wall time on warm runs. The math only pencils out
when test time significantly exceeds setup time, which is not the
case for Sencho today. Revisit if the suite grows past ~5 minutes
of pure test time.

.github/actions/start-app/action.yml

@@ -87,6 +87,33 @@ runs:
       shell: bash
       run: npx wait-on http://localhost:${{ inputs.port }}/api/health http://localhost:5173 --timeout 30000
 
-    - name: Install Playwright browsers
+    # Playwright bundles Chrome for Testing (~170MB) and FFmpeg. Downloading
+    # them on every E2E run is the single largest cost in this composite:
+    # observed ~4m15s cold vs ~10s warm when the path is cached. Cache the
+    # download under a key pinned to the @playwright/test version so a
+    # future bump invalidates automatically.
+    - name: Read Playwright version
+      id: playwright-version
       shell: bash
-      run: npx playwright install --with-deps chromium
+      run: |
+        version=$(node -p "require('./package.json').devDependencies['@playwright/test'].replace(/^[\^~]/, '')")
+        echo "version=$version" >> "$GITHUB_OUTPUT"
+
+    - name: Cache Playwright browsers
+      id: playwright-cache
+      uses: actions/cache@v4
+      with:
+        path: ~/.cache/ms-playwright
+        key: playwright-${{ runner.os }}-${{ steps.playwright-version.outputs.version }}
+
+    # OS deps are apt packages under /var, which actions/cache does not
+    # preserve, so they have to be reinstalled every run regardless of the
+    # browser cache state. It is fast (~10s).
+    - name: Install Playwright OS dependencies
+      shell: bash
+      run: npx playwright install-deps chromium
+
+    - name: Install Playwright browsers
+      if: steps.playwright-cache.outputs.cache-hit != 'true'
+      shell: bash
+      run: npx playwright install chromium