Feature request: Restart/revive a Remote Control session from the mobile app

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

When my `claude remote-control` server process dies while I'm away from the machine
(e.g., an extended network outage at home — the server exits after ~10 min offline),
the session shows offline in the mobile app and there is no recovery path without
physical or RDP access to the desktop. I'm currently traveling and locked out of my
session entirely.

### Proposed Solution

Either or both:
1. Auto-restart: have the Remote Control server retry/relaunch on network recovery
instead of exiting after ~10 minutes.
2. Remote revive: a lightweight always-on watchdog on the desktop that the mobile
app can signal to relaunch `claude remote-control`, so a dead session is
recoverable from the phone.

Related: #34119 (mobile-side disconnect/reconnect)

### Alternative Solutions

Task Scheduler / batch-file restart loops work but only cover process exit on the
machine side; they can't be triggered or verified from the mobile app.

### Priority

Critical - Blocking my work

### Feature Category

CLI commands and flags

### Use Case Example

_No response_

### Additional Context

_No response_

Start at the `claude remote-control` entry point and trace how the server handles prolonged network loss, process exit, and the mobile app's offline session state. Define whether recovery is automatic or signaled remotely, then verify that a session can be relaunched and its status confirmed from the mobile app after an outage.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.