[tests/ui] Finish Playwright CI matrix + remaining UI/integration test refactors

[mfittko]

Summary

Continue the UI/integration/E2E testing work: finish CI setup (browser matrix) and complete remaining DRY/refactor items that were left after the original UI testing issue was closed.

Source doc: docs/issues/planned/phase-5-ui-tests.md

Context: Issue #45 was closed, but phase-5-ui-tests.md still has open checklist items.

Tasks

• Review/refactor existing UI and integration tests for DRYness and maintainability
• Ensure key UI/integration flows and edge cases are covered
• Update GitHub Actions workflow to run Playwright via a browser matrix (Chromium, Firefox, WebKit)
• Keep Playwright caching efficient across the matrix
• Document UI/integration/E2E testing best practices

Acceptance Criteria

• Playwright E2E runs in CI on Chromium, Firefox, and WebKit
• UI/integration tests are maintainable and comprehensive
• Coverage remains ≥ 90%