I have found out something interesting: WinGet will let you know the URL from which it downloads the installer, so I decided to check whether the actual installer itself is at fault here, and it seems it is not, but I'm also not quite sure what's going on here.

For one, if you try to run the Discord installer WinGet downloads into %localappdata%\Packages\Microsoft.DesktopAppInstaller_8wekyb3d8bbwe\TempState\WinGet, that installer will refuse to work - as in will try to access C:\ProgramData\%USERNAME% - even if not executed from WinGet and executed with or without the same command-line parameters (WinGet runs it with /s).

If you download the EXE file through your browser instead or you decide to copy the EXE file WinGet downloaded to a new file, then both installer copies will work just fine. (EDIT: I double-checked with SHA256 hash checks and all hashes matched.)

Maybe this will help someone else to debug this? I am currently wandering through the source code and setting up for step-by-step debugging to find out what's going on, up until now I've found nothing useful other than this observation.