Jsgoddess and** featherlou**, a couple years ago, I took a few months of weekly private singing lessons. Not to be a performer of any kind. But I liked to sing and I just got tired of everyone shouting " please stop singing" when I opened my mouth.
I got what I wanted: I’m bearable now. 
I approach every new student with this same attitude. But I’ve met a number of people who simply could not be taught, and IMO you do have to have at least some “talent” in order to successfully improve.
When I say “talent,” I don’t mean the “wow - that person is an awesome singer!” kind. I just mean having a basic affinity with one’s own voice that allows for perception and (eventually) control.
Personality also plays a part. It takes the ability to step back from yourself and see your voice as a “thing” that you can work on, rather than just an extension of you. Some make that leap more easily than others.
There is hope! 
What’s kind of funny is that I’m fairly musical–I can play a handful of instruments, can read music and sight read, etc. And then I open my mouth.
I even do readings for poetry podcasts and get lots of compliments on my voice.
But you stick the two things together, and people start fleeing the room!
Sounds good to me - you lose a problem student
The parents are probably canny enough to realize that they are ‘buying fame’
- it is about time for a Punk Rock revival and your kid sounds fine for a Sid Vicious
I just wish I could think of a way in which you could benefit from the chain of events
- I have a glimmer, but it is inchoate - the idea is crystalizing - it just needs a name
… Double indemnity tuition - covers both opposing bases
- Debbie Harry meets Dame whatever - a sure winner.