Remote Control: expose the ~/.claude.json project registry as a picker in the session list

## Summary

Remote Control's session list shows only sessions that are already running. It should also expose the project registry Claude Code already keeps in `~/.claude.json`, so I can pick a project on my machine from the phone and start a session in it.

## Why this is a small ask

The data already exists. `~/.claude.json` records every directory Claude Code has run in — 21 entries on my machine, including nested ones like `C:\work\project-a\subproject`. The remote channel already exists. Only the picker is missing.

## Current workarounds, and why they fall short

Neither gives you tap-a-project-then-work:

1. **One named session per project** — `cd C:\work\project-a; claude --remote-control "Project A"`. Closest feel, but needs a separate terminal per project and burns a concurrent session on each.
2. **One wide session** — `claude remote-control --add-dir ...` from a parent, then navigate by typing `@`. Less setup, but you have to know the path already; there's nothing to browse.

Both require deciding what you want *before* leaving the desk, which defeats the point of picking something up from the phone.

## Prior art

The Codex app's remote view already does this — its Projects list is drawn from its own registry of directories you've worked in, and tapping one starts a session there.

## Suggested behavior

In the Remote tab, alongside the running-session list, show the projects from `~/.claude.json`. Tapping one starts a session with that directory as cwd, subject to the usual workspace-trust check.

## Environment

- Windows 11 Home Single Language (10.0.26200)
- Claude Code CLI, `claude remote-control` server mode
- Claude mobile app, connected and working

Start by inspecting the existing Remote tab session list and the Claude Code remote-control server path that creates sessions, then examine the project registry in ~/.claude.json, including nested directories. Done means registered projects appear alongside running sessions, tapping one starts a session with that directory as cwd, and the existing workspace-trust check still applies.

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