Desktop: "Connect new sessions to Remote Control" renders a three-state setting as a two-state toggle

### Environment
- Claude Desktop: 1.40609.0 (f65e38), built 2026-08-27
- Claude Code CLI on same host: 2.1.233
- macOS (Apple Silicon)

### Summary

`remoteControlAtStartup` is documented as taking three values — `true`, `false`, and `default`. The desktop app renders it as a two-position switch, so `default` and `false` are visually identical while behaving differently. A machine sitting at `default` shows the switch **off** while new sessions connect to Remote Control anyway.

This is not cosmetic: because the switch has no position meaning `default`, touching it converts an implicit `default` into an explicit value, and there is no way to get back to `default` from the desktop UI. The CLI's `/config` does offer `default` as a selectable value, so the two surfaces for one setting expose different choice sets, and only one of them can undo what the other does.

### Steps to reproduce

1. On a machine with no `remoteControlAtStartup` key set anywhere (fresh state = `default`), open **Settings > Claude Code**. The switch reads **off**.
2. Start a new Code session and open the session's Remote Control control.
→ Remote Control is **on**, and the session reports that it connected automatically at start. The session also appears in the Claude mobile app.
3. Toggle the setting **on, then off** — writing an explicit `false`.
4. Start another new session.
→ Remote Control is now **off**, and the session does not appear on mobile.

Steps 1 and 3 leave the switch in the same visual position with opposite behavior.

### Expected

The control distinguishes the three documented states, or the docs drop `default` for this surface. At minimum, entering an explicit value should be reversible from the same UI that set it.

### Secondary observation

After step 3, `remoteControlAtStartup` was present in **neither** `~/.claude/settings.json` nor `~/.claude.json`. The Remote Control docs name the first as where to set this, and describe the desktop switch as the same toggle surfaced outside the CLI. I did not locate where the desktop app does persist it, so this may be intended, but as written the docs imply a shared store that isn't shared.

Docs: https://code.claude.com/docs/en/remote-control#enable-remote-control-for-all-sessions

Start by reproducing the fresh-state behavior in Settings > Claude Code, then compare it with the CLI /config choices and inspect the documented locations ~/.claude/settings.json and ~/.claude.json. Done means the desktop control represents true, false, and default distinctly, allows an explicit value to return to default, and the Remote Control documentation accurately describes persistence.

Rédigé par le modèle d'indexation à partir du texte de l'issue.