ci(widget): run typecheck, unit tests, and Playwright in GitHub Actions

[Asaf-prog]

Problem

The widget has its own TypeScript typecheck, unit tests, and Playwright end-to-end tests, but these checks are not currently enforced by GitHub Actions.

The existing CI quality gate runs the Python-focused checks through make check, including linting, mypy, pytest, and generated-file validation. As a result, a PR can receive a green CI status even if the widget fails to compile or its browser tests are broken.

This is especially important for widget behavior involving:

• conversation switching
• streaming responses
• stale-conversation recovery
• localStorage state
• per-conversation message and usage state
• loading and typing indicators

These flows may only fail in TypeScript compilation or browser-level tests.

Proposed change

Extend the GitHub Actions workflow to run the widget checks on every pull request and push to the main branch.

The workflow should run:

1. Widget TypeScript typecheck
2. Widget unit tests
3. Playwright end-to-end tests

The exact commands should use the existing scripts defined by the widget project, for example:

npm run typecheck
npm run test:widget
npx playwright test

The workflow should also install the required Node.js dependencies and Playwright browser dependencies before running the tests.

Acceptance criteria

• GitHub Actions runs the widget TypeScript typecheck.
• GitHub Actions runs the widget unit tests.
• GitHub Actions runs the Playwright end-to-end tests.
• A failure in any widget check causes the PR CI check to fail.
• The checks run automatically on pull requests.
• The CI workflow uses the existing widget package scripts where available.
• Playwright browser installation is cached or configured efficiently where possible.

Expected result

A PR should not be considered fully green unless both the backend and widget test suites pass:

Python lint and typecheck     ✅
Python tests                  ✅
Widget typecheck              ✅
Widget unit tests             ✅
Playwright E2E tests          ✅