Now I’m really confused (phase cancellation with Audacity)..

I have never used Audacity but it may be that you have inverted both tracks without realising.
Try exporting the ‘karaoke’d’ track as a wav file at the same sample rate as the project and then re-import.
Then reverse the phase on a track.
Unfortunately Im not sure even this will work as the karaoke feature is likely to use heavy eq.
This will un-doubtably affect the instrumentation
in the mid range and probably prevent this area phasing.
Its a nice idea you have tho.