The police don't decide whether someone gets prosecuted for a crime, the state attorney does. The police only separate the suspect from the victim and if necessary isolate them from society for a cooling off period (which counts as time served if he is later convicted and sentenced), and investigate the facts for the prosecutor. In the course of these duties they need to be very by-the-book because if
either side objects to any of their actions, the accompanying facts cannot be used by the other side in their arguments (or in their defense, as the case may be).
Many police departments have developed special procedures, and even have special departments, to use when dealing with things such as sex crime, underage suspects, underage victims, and many of the other things that you claim the school is better at. One thing the school failed at is in the training of the principal in proper procedure and/or assuring that he would follow that procedure.
If it's not a violent crime or some kind of serious theft i wouldn't ever consider it necessary to get the cops involved.
What would be your definition of violent? A playground scuffle is violent to a certain degree, but it is not a police matter. Extortion is not violent, but it is a matter for the police. (And not all extortion is for money (or other material goodies) -- so the "serious theft" part does not entirely cover it.)
What works best is when all the authorities are on the same page, not fighting for turf schools against Child Welfare, against the police, etc. Together they should work out a policy (preferably before it's needed, as to the duties and responsibilities of everyone involved to ensure that it all is over as quickly and as painlessly as possible for the children involved.