'Run this program as an administrator' is also selected. Under the Compatibility section (accessible via right-clicking the swkotor.exe and selecting 'Properties:), I selected 'Disable Fullscreen Optimizations'.

None of the fixes mentioned online seemed to address this issue. Sometimes the main menu would display correctly, but then either the gameplay proper would still be a black screen or alt-tabbing out and in would then lead to a black screen. I could hear the button highlights when I moved the cursor around but could not see what I was clicking.

So I was having an issue where the main menu would be completely black and only my cursor was visible, while music would play.