Add Playwright test for Settings page provider save flow

[aitrace-dev]

Context

The Settings page is the first thing a new user touches on first run, and it's currently not E2E-tested. A single Playwright test covering "add a provider → save → reload → it's still there" would guard against a whole class of regressions.

Acceptance criteria

• New file packages/frontend/tests/settings.spec.ts
• Test flow: open Settings → fill the provider form with a fake API key → save → reload the page → verify the provider still appears
• No real network calls — use a fake key
• Passes locally with npm run test:e2e -w packages/frontend and in CI

Pointers

• packages/frontend/tests/settings.spec.ts (new)
• Existing patterns in packages/frontend/tests/chat.spec.ts

Difficulty

~1–2 hours. Great introduction to the Playwright + backend fixture setup.