Feature request: Live status indicators in session list (VSCode sidebar)

Summary

The VSCode session list (Local/Web picker) shows session titles but no indication of each session's current state. When running multiple sessions, the only way to know which one needs input, is actively working, or is idle is to click into each one.

Use case

Power users regularly run 3-5+ concurrent sessions. Without status indicators, managing them requires constant tab-switching to check state. This doesn't scale.

Proposed states

Visual indicators (icons, badges, or colored dots) next to each session title:

| State | Meaning | Example |
|-------|---------|---------|
| Needs input | Waiting on user (permission prompt, AskUserQuestion, tool approval) | Yellow dot or question mark icon |
| Working | Actively processing (agent running, tool executing, generating response) | Spinning/pulsing indicator |
| Idle / Complete | Finished last task, waiting for next prompt | No indicator or checkmark |

Why this matters

• Reduces context-switching overhead
• Prevents stale sessions from being forgotten (especially ones waiting on a permission prompt)
• Makes multi-session workflows viable at scale

Additional context

Related to #24811 (cross-session visibility) but intentionally narrower in scope — this is purely a UI improvement to the existing session list, not a coordination mechanism.

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