Desktop app: allow remote session *starts* on a machine, with a device picker in the mobile app

**Not a duplicate of #29006.** That request was about making Desktop-hosted sessions
*visible* from the phone, and it shipped as `Settings > Claude Code > Enable remote
control by default`. This asks for the opposite direction: starting a **new** session
on one of my machines **from** the phone, without touching that machine first.

## What I would like

1. **In Claude Desktop:** a per-machine checkbox, e.g. *"Allow remote session starts
on this device"*. Ticking it registers the machine as available, nothing else to
install or maintain.
2. **In the Claude mobile app:** a list of every machine where I ticked that box. Pick
a machine, pick one of its **known projects**, start a session there.

3. **In the sidebar, group remote sessions by machine.** Today every remote session
lands in the catch-all *Other* group, so sessions from my server, my desktop and my
bot PC are mixed into one list. Instead: one collapsible group per machine, named
the way that machine is named, with a small remote/device icon next to it - and the
same `+` button the local project groups already have, which starts a session on
that machine.

**No filesystem browsing from the phone, deliberately.** The phone should only offer
the project folders that machine already has - the ones in the Desktop app's own
project list, where I have started sessions before. Picking an arbitrary path from a
phone would be a permissions problem, and it isn't what I need: the folders I work in
are always the same handful. A machine with no projects yet simply offers nothing,
and I set it up once at the machine.

## Why

I run Claude Code on three machines: a game server, my desktop, and a bot PC. Right
now the only way to start a session on any of them from my phone is `claude
remote-control` in a terminal on that machine. So I first have to remote in (RDP /
Tailscale) to launch the very thing that is supposed to spare me from remoting in.
And if a machine rebooted and no server process is running, the phone is useless.

I don't want to run and maintain an autostart entry, service, or terminal window per
machine for this. A checkbox in the app I already have open is the whole point.

## What exists today

| | starts new sessions remotely | needs a process running |
|---|---|---|
| `Enable remote control by default` (Desktop) | no, only shares what I started locally | no |
| `claude remote-control` (CLI server mode) | yes, on demand | yes, a terminal on that machine |

The Desktop checkbox already covers half of this. Extending it to "this machine
accepts remote session starts" would close the gap, and the UI pattern is already
there: the sidebar groups local projects and gives each one a `+`. Doing the same
for each remote machine - name, device icon, `+` - would make it obvious, and it
beats today's single *Other* bucket even for people who never start a session
remotely.

Start by tracing the existing Desktop remote-control setting, the claude remote-control CLI server mode, and the mobile app's remote-session and project-selection entry points. Done means a machine can opt in, its known projects appear on mobile, new sessions can start there, and the sidebar groups remote sessions by machine with the existing + control.

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