fix(extension): detect stalled background worker

[huyua9]

Summary

• add a lightweight EXT_HEALTH_CHECK message handled by the extension background service worker
• make TabsController verify the background worker before task startup
• wrap background sendMessage calls with a short timeout and raise an actionable recovery error instead of returning null

Why

Issue #452 reports the side panel becoming unresponsive when Chrome MV3 leaves the extension service worker in a stuck/inactive state. The current tab-control path logs chrome.runtime.sendMessage failures and returns null, which can turn the real service-worker problem into a less helpful null dereference during task startup.

This keeps the service worker stateless/restartable, but gives the side panel a deterministic health probe and a clear failure message that points to the maintainer-provided recovery steps from #452.

This is intentionally scoped to the background/TabsController path and does not touch the sidepanel UI copy, so it complements the existing startup-guidance PR without overlapping its file scope.

Validation

• npm ci --ignore-scripts --no-audit --no-fund (npm 10.9.4 reports the repo's npm ^11.6.3 engine warning)
• npm run postinstall --workspace=@page-agent/ext
• npm run typecheck
• npx eslint packages/extension/src/agent/TabsController.ts packages/extension/src/entrypoints/background.ts
• git diff --check
• npm run build

[CLAassistant]

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