Bug: Notification badge count persists after all notifications cleared (Windows)

## Description

The Copilot desktop app (Windows) shows a notification badge count on the system tray/taskbar icon even when there are no outstanding notifications, pending plan approvals, or inbox entries.

## Steps to Reproduce

1. Use the Copilot desktop app normally
2. Clear/read all notifications and pending items
3. Observe the app icon still shows a numeric badge (e.g. '1' or '2')

## Expected Behavior

The badge count should be 0 (no badge shown) when there are no unread notifications or pending items.

## Actual Behavior

The badge persists with a stale count. It does not decrement to zero even after:
- Viewing all notifications
- Confirming inbox is empty (0 entries in session database)
- Reinstalling the app
- Clearing all caches
- App self-updating the database

## Environment

- OS: Windows
- Copilot App Version: 1.0.78-2
- The issue persists across reinstalls and cache clears

## Additional Context

This appears to be a state management issue where the badge count is not properly synced with the actual notification state.

No files, tests, or entry points are identified. Reproduce the stale badge on Windows, then trace how notification and pending-item state updates the tray or taskbar count; done means the badge reaches zero when notifications and pending items are cleared, including after reinstall and cache-clear scenarios.

Written by the indexing model from the issue text.