Agree with your entire post except the reoccuring question: "Why don't they just call the cops?"
If they call the cops much of the show's premise is over since this is a martial arts combat and bullying themed show, not a psychological thriller or romantic intrigue etc.
But from a "realism" standpoint, it never bothered me because I never even thought of that as a possibility for kids and men whose pride/toughness is challenged. In my lived experience and from what I've seen of others, when the average kid is bullied or the subject of violence, they are too humiliated to tell an adult. Everyone will know you called the cops, akin to everyone knowing you called Mommy and Daddy like Eli endured before he flipped the script.
The culture growing up was to handle it yourself. Also, many kids get caught up in thinking high school is everything. And don't realize there are real resources outside of high school but much of it is about pride and shame. It's why some kids who get bullied don't report it and end up committing suicide (like those who spawned the "it gets better" campaign). Some others, like many mixed martial artists, say they got started in it just like these kids--to take matters into their own hands and learn how to fight.
As for men, from what I've observed and experienced, calling the cops to resolve your fights and challenges is basically admitting you lost the pissing match, so to speak. I easily believe they would settle martial arts rivalries via martial arts and not law enforcement.