Expose assignment failure information via health check API

Since specialization is async, App Service might incorrectly assume instance is specialized successfully. Expose this failure so App Service can take remedial action instead of leaving the borken instance alive.

No files or tests are named. Start by locating the asynchronous specialization assignment flow and the health-check API entry point in the C# host; trace how assignment failures are represented and surfaced. Done means the health check exposes specialization failure information clearly enough for App Service to take remedial action.

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