test: move PW_CLOCK handling into test harness

[dgozman]

Drive-by: remove mode.startsWith('service') checks after the service mode removal.

[github-actions]

Test results for "MCP"

3 failed
❌ [chrome] › mcp/annotate.spec.ts:57 › should capture multiple screenshots in one annotation @mcp-windows-latest-chrome
❌ [chrome] › mcp/annotate.spec.ts:110 › should abort annotation when last screenshot is removed @mcp-windows-latest-chrome
❌ [chrome] › mcp/annotate.spec.ts:137 › should abort MCP annotation when last screenshot is removed @mcp-windows-latest-chrome

7263 passed, 1119 skipped

---

Merge workflow run.

[github-actions]

Test results for "tests 1"

7 flaky

⚠️ [chromium-library] › library/beforeunload.spec.ts:130 › should support dismissing the dialog multiple times `@chromium-ubuntu-22.04-arm-node20`
⚠️ [chromium-library] › library/video.spec.ts:719 › screencast › should work with video+trace `@chromium-ubuntu-22.04-arm-node20`
⚠️ [chromium-page] › page/page-request-continue.spec.ts:756 › propagate headers cross origin redirect after interception `@chromium-ubuntu-22.04-arm-node20`
⚠️ [chromium-library] › library/video.spec.ts:682 › screencast › should capture full viewport on hidpi `@chromium-ubuntu-22.04-node20`
⚠️ [chromium-library] › library/chromium/chromium.spec.ts:177 › serviceWorker(), and fromServiceWorker() work `@chromium-ubuntu-22.04-node22`
⚠️ [chromium-library] › library/chromium/chromium.spec.ts:434 › should produce network events, routing, and annotations for Service Worker (advanced) `@chromium-ubuntu-22.04-node22`
⚠️ [chromium-library] › library/video.spec.ts:476 › screencast › should capture static page in persistent context @smoke `@chromium-ubuntu-22.04-node22`

39485 passed, 774 skipped

---

Merge workflow run.