Add sync health checks and doctor output

Context

Future work from docs/superpowers/specs/2026-05-24-local-libsql-migration-design.md.

The local libsql migration and embedded replica/sync mode are prerequisites. This issue should expose direct status signals rather than inferring health from side effects.

Scope

• Add explicit health probes for configured cloud-backed storage modes.
• Report sync/replica status through doctor and runtime health surfaces.
• Define thresholds for stale sync, authentication failures, and unreachable remote databases.
• Ensure errors propagate to logs and operator-visible checks.

Acceptance Criteria

• Local-only agents do not show irrelevant cloud warnings.
• Cloud-backed agents expose direct sync status and last-error details.
• Doctor output includes actionable remediation without leaking secrets.
• Tests cover healthy, stale, auth-failed, and unreachable states.

Out Of Scope

• Designing cloud configuration.
• Implementing backup/restore UX.
• Migrating from rusqlite to local libsql.

Start with docs/superpowers/specs/2026-05-24-local-libsql-migration-design.md, then locate the doctor and runtime health surfaces that handle cloud-backed storage. Define how healthy, stale, authentication-failed, and unreachable states appear, and run or add tests covering those states, local-only warning suppression, actionable remediation, and secret-free output.