refactor(analytics): standardize logging and prevent resource leaks

[victoriacheng15]

Summary

Enhances the analytics engine by enforcing standardized observability practices and ensuring proper resource lifecycle management. These adjustments eliminate unformatted standard output and prevent potential connection leaks during service shutdown, aligning with our operational excellence standards.

List of Changes

• Standardized initialization error logging to utilize the core telemetry framework, ensuring all startup logs are properly formatted and captured.
• Implemented explicit connection closure for the secret provider to guarantee a graceful shutdown and prevent resource exhaustion at scale.

Verification

• Run unit tests for the analytics service to ensure initialization logic is sound.
• Deploy the service locally and manually verify that graceful shutdown correctly closes the secret store without connection leaks.