I would expect it to be used in the majority of cases to be so considered – not the minority. In many schools, I could have smoked; played truant; swore at teachers; carved my name on desks; wandered where and when I wanted; in general do as I please – all without fear of getting the cane. Can it truly be said that, in such schools, the cane was the punishment for any of those offences?
Probably not – but you’re confusing two different things here. What you seem to be describing are schools where the cane was never used, or was used extremely rarely (despite apparently claiming it was supposed to be used?) There’s a lot of room between those schools, and schools where it was used reasonably often, but still only in a minority of cases.