I'm not able to locate any context in which the biblical "rod" is used to hit people. It is never imagined as a paddle. That idea was brought in by Christians.

As with Moses' staff or the many "staffs" and "rods" on view throughout the OT, we're thinking of a magical object that indicates the power of a rightful ruler.

The context of Proverbs 13:24 is not ordinary childrearing. It's about Israelite kings and the transmission of the wisdom of rulership. We're thinking about succession and suitable heirs to the throne.

In the OT narratives, sons are problematic. In the usual scene, they try to overthrow the father-king. In that context, Prov 13:24 says to teach kingly wisdom to eventual heirs since they are children.