[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`

Start by reading the Remote Control startup path around `remoteControlAtStartup`, the `/rc` flow, and the `claude remote-control` server entry point; the payload only names `~/.claude/settings.json` as configuration. Also review related issues #34255 and #69543 for failure modes. Done means sessions connected before an app update/relaunch reconnect without manual action, and the iOS app can see the Desktop-managed server as described.

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