Playwright test for Run trust loop on /demo #40

Summary

Browser tests only assert the trust-loop button is visible. API coverage lives in tests/integration/test_demo.py. The public demo story is cite FAQ → replay → inbox; that UI path is untested.

Context

• tests/browser/test_critical_paths.py (test_demo_mic_control_visible)
• public/demo/index.html / public/demo/app.js
• Distinct from Browser test for microphone-triggered barge-in #31 (barge-in) and Playwright 375px smoke for public landing and /demo #20 (375px smoke)

Acceptance criteria

• Click Run trust loop with DEMO_ENABLED=true
• Assert citations, inspect/replay links, and a success status in #out-status
• Stable in CI with mock providers