refactor(startup): inject sync service dependencies

[pablontiv]

Related to #48 phase 1c.3c.

Summary

• Remove mutable startup sync, diagnostics, and recover globals.
• Share one per-root sync service between startup coordination and recovery.
• Keep diagnostics timing local to each execution.
• Make injectable construction nil-safe with production defaults.

Validation

• just ci passed with 85.3% total coverage.
• Full race evidence applies to the exact equivalent patch; focused race checks passed before the equivalent rebase.
• Candidate review accepted after nil-safety hardening.