Technology
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Dang, that’s a really cool phenomenon, thanks for sharing!
If the common wire is broken, maybe depending on how the headphones are wired you will hear the difference between left and right instead?
This might be it. Depending on how the sound is mixed the voice might cancel itself out since it's usually similar between left and right and the instruments are not. So they will not experience the same interference
This is the most likely answer. Leading vocals (and instruments) are normally centered, i.e. the same level on both channels, while background instruments are louder on one channel.
With the ground wire gone, you don't have a stereo signal left-ground / right-ground anymore, but a mono signal left-right, i.e. you only hear the difference between the two channels. That cancels out any centered vocals and instruments.
The volume will also likely be much lower, as the signal has to travel through twice the resistance (two speakers) at probably roughly the same voltage.
This actually used to be a nice trick to get a pseudo-instrumental version of any song - just subtract the channels from each other. You'll get a mono version of the song with only the background music. Not sure how it's done today, there's probably a better solution now.
Your question is worded very oddly, but to attempt to answer it, headphones (most) operate off of an analog signal from a DAC (digital to analog converter). In any analog setup you're going to have some amount of signal transmission or line noise unless the two transmission lines are entirely shielded or entirely separated.
This seems most plausible. OP, if you have a multimeter you could try to trace which TRS terminal has continuity with the damaged wire (if it's exposed) and see which signal it's supposed to carry.