fix: keep playwright out of package.json

It slipped in from a local UI check. The Docker build installs devDependencies,
and playwright's postinstall would try to download browsers there - the same
class of failure as the tsx/esbuild one.

package.json

@@ -36,7 +36,6 @@
     "@types/pg": "^8.11.10",
     "@types/react": "^19.0.0",
     "@types/react-dom": "^19.0.0",
-    "playwright": "^1.62.1",
     "typescript": "^5.7.2"
   }
 }