Chosaych Shivto may not mean sparing the actual rod. The shevatim were marked by a degel on a stick that went before each tribe. Maybe this refers to the leadership before each shevet, and maybe it means, that a parent who withholds proper PARENTING/leadership when his child does wrong, is a parent who does not love his child enough to correct him.
OK I am stretching a bit, but I like my interpretation.