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.

索引モデルが issue の本文から書いたものです。