Command Palette gets stuck after PowerToys starts and PowerToys itself runs in task manager

### Microsoft PowerToys version

0.99.1

### Installation method

WinGet

### Area(s) with issue?

Command Palette

### Steps to reproduce

1. Install PowerToys 0.99.1.
2. Enable Command Palette in PowerToys.
3. Enable PowerToys Run in PowerToys.
4. Set PowerToys to run at startup.
5. Restart Windows.
6. After login, wait for PowerToys to start.
7. Try opening Command Palette with the shortcut.
8. Check Task Manager and the system tray.

### ✔️ Expected Behavior

PowerToys should start normally after Windows login.
The PowerToys tray icon should appear.
Command Palette should start once and open immediately when using the shortcut.
Command Palette should not stay stuck as a background process only.
PowerToys Run and Command Palette should both remain usable after startup.

### ❌ Actual Behavior

PowerToys can start and appear in task manager, but the tray icon doent always appear.
Command Palette can also appear in task manager, but it doesnt reliably open from the shortcut.
Sometimes Command palette process is running, but it is not usable.
Sometimes I need to manually kill Microsoft.CmdPal.UI.exe and relaunch Command Palette before it works.
The Command Palette startup app is disabled in Windows Startup apps, so it is not being launched separately from there.
PowerToys itself is started from a Task Scheduler task using this path:
C:\Users\\AppData\Local\PowerToys\PowerToys.exe

That path exists and PowerToys can run from it.
Other PowerToys modules work without issues, but Command Palette is unstable at computer restart.

Command Palette logs show:
Failed to save:
C:\Users\\AppData\Local\Packages\Microsoft.CommandPalette_8wekyb3d8bbwe\LocalState\settings.json
The process cannot access the file because it is being used by another process.

### Upload Bug Report ZIP-file

_No response_

### Additional Information

I already tried uninstalling PowerToys, deleting the old local PowerToys folders, deleting the Command Palette package state folder, and reinstalling PowerToys fresh with winget.
After reinstall, the old corrupted state are gone, but this startup and command palette stuck behavior happens again.

### Other Software

Windows version:
Microsoft Windows 10.0.26200
Build shown in CMD:
10.0.26200.8246

PowerToys version:
0.99.1

Command Palette app version:
0.10.11181.0

Command Palette runtime:
.NET 9.0.15

System language:
en-GB

Running as admin:
No

Start by reproducing the restart scenario with PowerToys.exe launched by Task Scheduler, Command Palette enabled, and PowerToys Run enabled. Inspect Microsoft.CmdPal.UI.exe behavior alongside the reported settings.json file-lock error. Done means the tray icon appears after login, Command Palette opens reliably from its shortcut, and both modules remain usable without manually killing the process.

Written by the indexing model from the issue text.