I’m not quite sure if we have a disagreement, or just a disagreement about vocabulary:
My contention is that the police do law enforcement, they don’t do trials and they don’t do punishments(at least by design). They do have a lot of room for ‘informal’ punishment(“You can be the rap; but you can’t beat the ride”, rough driving with handcuffed people in the car, various other applications of force, petty and grave, up to and including just gunning people down). These, however, are all illegitimate functions; bullying at best, assault or murder at worst.
Unless self defense or public safety requires it, they are supposed to either issue a citation or bring you in with as little fuss as possible, at which point the judicial branch takes over, assesses your guilt, and selects a punishment.
Nobody denies that cops have a great deal of practical leeway to punish people off the record; but my contention is that doing so is contrary to their mission, illegitimate, and correctly perceived by the public as bullying, coercion, or outright threat of dangerous or fatal violence; which breeds either cringing avoidance or rebellion; but not respect.