Desktop: Chat/Cowork mode picker only appears on cold app launch

### What happens

The Chat/Cowork mode toggle is visible on the empty composer when the desktop app is launched. Once a conversation has started, the toggle disappears and that conversation is permanently locked to the mode it began in.

`Cmd+N` (new conversation) returns to the picker, but there is no way to switch an in-flight conversation.

### Expected

Mode should be switchable mid-conversation, or the picker should remain reachable without starting a new conversation.

### Impact

Choosing the wrong mode means abandoning the thread and re-establishing context in a fresh one.

### Environment

Claude desktop app, macOS.

### Related

#50341 — with Cmd+1/Cmd+2/Cmd+3 not working, a hidden picker leaves no switching path at all.

Start by tracing the desktop app's empty composer, mode picker, conversation state, and Cmd+N flow. Confirm how Chat/Cowork is selected at launch and verify that the completed behavior allows mode selection during an existing conversation or keeps the picker reachable without abandoning the thread.

Written by the indexing model from the issue text.