I can see the red clock in the home screen status bar after I close KORG Gadget, even though Background Audio mode is off. Why?

This is the expected behavior as of v4.5.4 to fix other issues.
The app may continue to run in the background, but the processing cost is very low because the synths and effects are deactivated when the app is not in use. Also, please note that no audio will ever be recorded without notifying you while the app is in the background.