This is old school now, but in old episodes of "Leave it to Beaver", "The Andy Griffith Show", and "Brady Bunch", the parents try to intervene and mediate (by talking to bully's parents who are of course bully enablers). In all instances, Beaver/Opie/Peter resolve by hitting the cowardly bully. I was taller and had 3 older brothers. Nobody messed w me, but in 4th grade once I told the 5th grade bully I was going to put a stop to it (he had the 2 "seconds", classic), about 50 people gathered around, I popped him good and he started crying, saying I'd have to pay the doctor bill (case closed). My teacher (on yard duty) just kinda smiled.
I was the "protector", but one of the little friends I protected had a smart mouth and could almost give better than he got. One day I decided to not protect him, so a kind of justice could be served.
One of my boys got popped by an older kid (whose Dad was in San Quentin), so I had his older Boy Scout friends "goon" the kid, case also solved. Glad you advocate "letting the kids work it out". As you say, when the school intervenes and tries to "mediate" it just empowers the bully. I guess the school gets to wash its hands and feels they've administered "therapy", but there's nothing like a little "knuckle treatment" (often simply a shove back). Am I running afoul of Gandhi and Dr. King here?