[FEATURE] Desktop app should keep Remote Control connected automatically across app updates (Codex-style always-on)

Problem

Remote Control in the Desktop app does not stay available on its own. With remoteControlAtStartup: true set, new sessions connect fine — but:

• Sessions that are open while the app updates lose the Remote Control link and never re-run the startup auto-connect. Each one needs a manual /rc or a close-and-reopen.
• After some idle time or an update, the iOS app shows nothing reachable until a session on the Mac is manually re-armed.

Related: #34255 (silent drop, no recovery), #69543 (auto-connect dies after overnight idle).

Feature request

Make the Desktop app keep Remote Control connected on its own, the way the Codex app does:

1. Re-run the auto-connect flow after an app update / relaunch for sessions that were connected before, not just fresh sessions.
2. Ideally: an option in the Desktop app to run a background Remote Control server (equivalent to claude remote-control server mode) managed by the app itself — always reachable from the mobile app, no terminal, no per-session step.

Workaround today

A user-managed launchd agent running claude remote-control with KeepAlive works well (server survives updates because launchd restarts it with the new binary), but this shouldn't require hand-rolled launchd plists.

Environment

• Claude Code 2.1.246, macOS (Darwin 25.5.0), Desktop app + iOS app
• remoteControlAtStartup: true in ~/.claude/settings.json