It's not an issue with BGInfo. It's an issue with Windows 11. If you look at the referenced Registry key (\SOFTWARE\Microsoft\Windows NT\CurrentVersion\ProductName), you'll see that Windows 11 thinks it's Windows 10.
I read that MS says they never intended these values to be used by anyone/anything so they don't bother updating. Stoopid MS.
I'm looking for a suitable alternate now so if anyone has ideas, I'd like to hear.