Add Playwright E2E test for critical path

[gauthierpiarrette]

Problem
There are zero end-to-end tests. Unit tests cover individual modules but don't verify the full user journey works.

Proposed solution
Add one Playwright test covering the happy path:

1. Start mcpplay demo as a subprocess
2. Wait for the server to be ready on localhost:8321
3. Open the browser to the UI
4. Verify the tool list loads (sidebar shows tools)
5. Click on a tool (e.g., echo)
6. Fill in the form (type a message)
7. Click Execute
8. Verify the result appears in the result panel
9. Verify a timeline entry was created

This single test covers: CLI startup, proxy connection, WebSocket communication, form rendering, tool execution, result display, and timeline recording.

Alternatives considered
Cypress or Puppeteer instead of Playwright. Playwright is the standard for modern E2E testing.