It is you who do not understand, either the language or the hermeneutics of these passages in Solomon's Proverbs. The Scripture is revealed by God and is His non-negotiable instruction. Normal interpretation is basic. This passage is basic. Literal interpretation is normal. Literal interpretation contains both literal and figurative-literal language. Figurative and/or allegorical interpretation (such as the one you propose) is not normal. The passage referred to (Prov. 13:24) has only one primary interpretation, only one meaning. it has only one sense. The rule is that when the plain sense of Scripture makes common sense, seek no other sense.
The grammatical sense of the word "rod" is "shebet" in Hebrew, and is simple -- it means a stick. This spare-the-rod passage is in literal language and is to be translated and interpreted literally. It is taken as a direct order to a real parent from The God in clear terms of dealing with disobedience:
"He that spareth his rod hateth his son: but he that loveth him chasteneth him betimes."
"Betimes" means "early" -- immediately after the infraction, while the lesson may still be learned.
There's more to this, but your application is not a correct interpretation.
Here's another passage, with a literal interpretation but with figurative-literal language:
"In the mouth of the foolish is a rod of pride: but the lips of the wise shall preserve them." (Prov. 14:3)
And the very word you want to quote is indeed a scepter a mark of authority. No doubt you would apply the same logic to the parsing of ‘Thy rod and Thy staff they comfort me.” Nice try but....not