test: cover common logging example controller

[brentthomas248]

Summary

• adds controller unit coverage for the common logging example controller
• verifies the happy path logs through mocked LoggerService and RequestContextService dependencies
• verifies the error path records structured error details before surfacing the thrown error

Closes #1086

Verification

• npm test --prefix backend -- example.controller.spec.ts
• npm test --prefix backend -- logger.service.spec.ts example.controller.spec.ts
• git diff --check HEAD~1..HEAD -- backend/src/common/examples/example.controller.spec.ts
• cd backend && npx --no-install eslint src/common/examples/example.controller.spec.ts