I confine myself to the specific question you ask, eschewing for the nonce to lay out an exposition of corporal punishing in general (which, no doubt, someone else will introduce momentarily).
“Rod” appears nine times in the Book of Proverbs. I first submit that six of these expressly address the application of the rod for purposes of the inculcation of wisdom to a child (Prov. 10:13, Prov. 13:24, Prov. 22:15, Prov. 23:13, Prov. 23:14, Prov. 26:3, and Prov. 29:15).
Prov. 10:13 and especially Prov. 26:3 speak of using the rod to impose guidance or motivation to a fool, and these might reasonably speak to the condition of a child insofar as children below a certain age lack the wisdom which the other proverbs name as the goal for which a parent applies a rod to such a child. Proverbs 22:15 puts it this way: Foolishness os bound up in the heart of a child."
In all these six (or, possibly, eight) proverbs, the rod is the instrument recommended in order to achieve the correction that leads to wisdom.
The Proverbs that mention the rod also observe that wisdom may also arise from words (Prov. 10:13, Prov. 14:3), especially a rebuke Prov. 29:15). So there is no necessary requirement that a rod be used if words will do.
But, the pessimistic evaluation in the Proverbs - viz. that children are foolish - is the starting point for the recommendation that the rod be used in order to successfully achieve the correction the parent seeks.
Is the rod simply a synechdoche for the more abstract notion of correction? And, if so, does this invite the speculation which you’re inquiring about, namely that correction is what a parent may seek apart from spanking?
As to the last speculation, the Proverbs emphatically say “No!” Correction apart from application of the rod may, indeed, be a desirable stage in the evolution of a child’s character as he progresses from fool to sage. It’s a no brainer that wisdom is best achieved by the reception of the words of those who are wiser than the student of wisdom. That said, the proverbs named above insist that a child’s heart is so infused with folly that the first steps out of that deplorable condition are achieved only by judicious application of the rod.