Sync subagent multi-agent mode with root

[shijie-oai]

Summary

• Resolve the root thread's selected multi-agent mode whenever a thread-spawn subagent builds a new turn
• Keep per-agent effective-mode calculation based on the subagent's own model, feature state, and source
• Preserve the subagent's persisted selection as a fallback when the root thread is unavailable

Why

Previously, subagents copied the parent mode when they were spawned and restored their own last effective mode after residency eviction. Changing the root mode could therefore leave loaded and revived subagents operating under different delegation hints.

This change keeps the root thread authoritative for subsequent turns without coupling prompting policy to V2 residency.

Validation

• just test -p codex-core multi_agent_mode
• just fix -p codex-core
• just fmt

Stacked on #28792.

[shijie-oai]

Superseded by #28792. The retained behavior is the narrower spawn-time inheritance: V2 passes the invoking turn's effective multi-agent mode into the new child, and the child keeps that mode after spawn. The per-message mailbox synchronization from this draft was intentionally dropped.