Expose remote control status in statusline JSON payload

Problem

The custom statusline script receives comprehensive JSON session data (model info, cost, context window usage, session ID, vim mode, etc.), but there is no field indicating whether the current session has remote control enabled.

Users who customize their status line have no way to visually indicate whether their session is accessible remotely.

Suggested approach

Add a field to the statusline JSON payload, e.g.:

{
  "remote_control": {
    "active": true
  }
}

This would allow custom statusline scripts to display an indicator (e.g., an icon or label) when remote control is active.

Use case

When working with multiple Claude Code sessions (some local-only, some with remote control), it's useful to see at a glance which sessions are remotely accessible — especially for security awareness.