feat: Add Playwright automated testing for authentication flow

- Install Playwright for automated browser testing
- Create comprehensive auth flow test script with screenshots
- Verify authentication fixes prevent flashing and redirect loops
- Generate visual proof that unified auth system works correctly

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <noreply@anthropic.com>

package.json

@@ -49,6 +49,7 @@
     "cheerio": "^1.1.0",
     "dotenv": "^17.1.0",
     "node-cron": "^4.2.0",
+    "playwright": "^1.54.1",
     "qrcode": "^1.5.4",
     "ramda": "^0.31.3",
     "react": "^19.1.0",