Let BackgroundTaskObserver observe shutdown cancellation

[Widthdom]

Found while dogfooding the locally built cdidx on branch codex/search-audit-issues-20260619.

Duplicate check: no open issue found for BackgroundTaskObserver shutdown cancellation Task.Run CancellationToken.None.

Evidence:

• BackgroundTaskObserver observes background work via Task.Run using a non-shutdown cancellation token.

Problem:
Observer tasks should participate in process shutdown so cancellation, disposal, and test teardown do not depend on fire-and-forget scheduling behavior.

Acceptance criteria:

• Thread an appropriate shutdown token into background observer scheduling.
• Preserve existing exception observation behavior.
• Add tests or teardown coverage proving observer work does not outlive shutdown unexpectedly.

[Widthdom]

Work started on branch fix-issue3658-3760-3705-3758-3743.