But that's the point: for corporal punishment to exist, you already have to have authority over someone by other means. "Comply and let me inflict pain on you"

But that's the point: for corporal punishment to exist, you already have to have authority over someone by other means. "Comply and let me inflict pain on you" seems very weird when it's equally true that "comply and be bored for a while" is right there as an option.