Taskbar badge does not show until the app window closed

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### Describe the bug

I am using `Windows.UI.Notification.BadgeUpdateManager` api to add a badge to taskbar icon for a winui3 app, the badge does not appear in the taskbar icon until the window is closed. And it will disappear after it launches. In short, it seems that a winui3 app cannot show the badge when the app window is running.

**For UWP app however, badge will appear as soon as the function is called.**

### Steps to reproduce the bug

1. Use following code
```cpp
inline auto MakeBadgeNotification(wchar_t const* glyph)
{
auto xml = winrt::Windows::UI::Notifications::BadgeUpdateManager::GetTemplateContent(
winrt::Windows::UI::Notifications::BadgeTemplateType::BadgeGlyph
);
xml.SelectSingleNode(L"/badge")
.as()
.SetAttribute(L"value", glyph);
return winrt::Windows::UI::Notifications::BadgeNotification{ xml };
}

inline auto MakeBadgeNotification(int value)
{
auto xml = winrt::Windows::UI::Notifications::BadgeUpdateManager::GetTemplateContent(
winrt::Windows::UI::Notifications::BadgeTemplateType::BadgeNumber
);
xml.SelectSingleNode(L"/badge")
.as()
.SetAttribute(L"value", winrt::to_hstring(value));
return winrt::Windows::UI::Notifications::BadgeNotification{ xml };
}

winrt::Windows::UI::Notifications::BadgeUpdateManager::CreateBadgeUpdaterForApplication().Update(MakeBadgeNotification(1)); //number badge
winrt::Windows::UI::Notifications::BadgeUpdateManager::CreateBadgeUpdaterForApplication().Update(MakeBadgeNotification(L"alert")); //glyph badge
```
2. Badge only appears in start menu, not in taskbar
3. Close the app, the taskbar pinned icon now has badge.

### Expected behavior

_No response_

### Screenshots

This is Winui3 app

After closing the window:

This is UWP app

### NuGet package version

Windows App SDK 1.4.2: 1.4.231008000

### Packaging type

Packaged (MSIX)

### Windows version

Windows 10 version 1809 (17763, October 2018 Update)

### IDE

_No response_

### Additional context

_No response_

Start by reproducing the packaged WinUI 3 behavior with the shown MakeBadgeNotification code and BadgeUpdateManager APIs on the reported Windows and Windows App SDK versions. Compare taskbar and Start menu behavior with the UWP case, then trace the Windows App SDK notification integration to identify where live taskbar updates diverge. Done means badges appear while the WinUI 3 window is running and do not disappear after launch.

Written by the indexing model from the issue text.