Yeah, that's a big one. Remember we don't have a health system in general. Most of us have private health insurance from employers, retired people have Medicare, retired military have Tricare from the VA and so on.
But you do bring up a valid point. It's always been a question. Why should *I* have to pay more for insurance or taxes because idiots engage in extreme sports and get hurt? What about people who smoke? What about the overweight? What about those like myself, diabetes and heart disease? I just got out of hospital - the bill to my insurance company for a four hour surgery on my heart was $169,000 USD. I didn't have to pay it, my insurance did (and everyone who pays into that insurance presumably). Do we assign blame and costs based on how much of my illness is genetic and how much is my fault from not taking care of myself better?
The bottom line is this. People get hurt for lots of reasons. Preventable and otherwise. As societies, we choose to deal with those costs in terms of insured and uninsured, taxpaying citizens and not. But we don't separate them out based on blame such as "was taught well by a qualified black belt" and "was taught poorly by a fake black belt."
If you wish to pursue that sort of thing, that's civil law - a tort. You can sue someone for having taught you or your child poorly or who lied about their credentials. You can't have them arrested. It's simply not a crime.