The Rod the Biblw speaks of is a sign of authority not an instrument of beating. One is to use their authority and wisdom to disciple i.e. to teach or instruct
The word translated “beat” in Proverbs 23:13-14 is Hebrew “nakah”, and it definitely means to strike physically, whether lightly or severely. So yes, the rod here (from Hebrew “shebet”, a small tree branch) is most assuredly an instrument for corporal punishment.
Wrong and liberal, for that is only one use of the word shêbeṭ, , while it basically means to
to branch off; a scion, i.e. (literally) a stick (for punishing, writing, fighting, ruling, walking, etc.) or (figuratively) a clan:—× correction, dart, rod, sceptre, staff, tribe. - Strong's, H7626 )
And thus it often speaks of an instrument used by authority, including in discipline, the very thing you deny!
And if a man smite his servant, or his maid, with a rod , H7626 and he die under his hand; he shall be surely punished. - Exo 21:20
He did not smite the slave with a mere claim of authority, but an instrument used by authority.
John Gill provides some insight from Jewish commentary (for what its worth):
Exodus 21:20: And if a man smite his servant or his maid with a rod,.... A Canaanitish servant or maid, as the Targum of Jonathan, and so Jarchi; and that only with a rod for the correction of them, and not with a sword or any such destroying weapon, which would seem as though he intended to kill, yet nevertheless:
and he die under his hand; immediately, while he is smiting or beating him or her, on the same day, as the above Targum interprets it:
he shall be surely punished; or condemned to the punishment of being slain with the sword, as the said Targum and Jarchi explain it: this law was made to deter masters from using severity and cruelty towards their servants.
2Sa 7:14: I will be his father, and he shall be my son. If he commit iniquity, I will chasten him with the rod H7626 of men, and with the stripes of the children of men:
God is not chastening by the authority itself, but with instrumentation of it.
Psa 23:4 Yea, though I walk through the valley of the shadow of death, I will fear no evil: for thou art with me; thy rod H7626 and thy staff they comfort me.
Phillip Keller in "A Shepherd Looks At Psalm 23" relates ,
Each shepherd boy, from the time he first starts to tend his father's flock, takes special pride in the selection of a rod and staff exactly suited to his own size and strength...The rod was what he relied on to safeguard both himself and his flock in danger. And it was, furthermore, the instrument he used to discipline and correct any wayward sheep that insisted on wandering away...If the shepherd saw a sheep wandering away from its own, or approaching poisonous weeds, or getting too close to danger of one sort or another, the club would go whistling through the air to send the wayward animal scurrying back to the bunch...Finally the shepherd's rod is an instrument of protection both for himself and his sheep when they are in danger. It is used both as a defense and a deterrent against anything that would attack. ...
Pro 10:13: In the lips of him that hath understanding wisdom is found: but a rod H7626 is for the back of him that is void of understanding.
Obviously the rod is a a literal instrument and used for correction, and thus it may also refer to what authority can do by other means of correction.
Pro 13:24: He that spareth his rod H7626 hateth his son: but he that loveth him chasteneth him betimes.
The literal rod is used for chastening, and to not use it if and as needed is actually hateful in effect, despite the emotion involved.
Pro 22:15: Foolishness is bound in the heart of a child; but the rod H7626 of correction shall drive it far from him.
Thus the rod represents physical chastisement.
Pro 23:13,14 Withhold not correction from the child: for if thou beatest him with the rod , H7626 he shall not die. Thou shalt beat him with the rod , H7626 and shalt deliver his soul from hell.
Once again obviously the rod is a a literal instrument and used for correction.
Pro 26:3 A whip for the horse, a bridle for the ass, and a rod H7626 for the fool's back.
Thus the rod is like a whip for the horse,
Pro 29:15 The rod H7626 and reproof give wisdom: but a child left to himself bringeth his mother to shame.
Thus the just and wise use of physicality correction is part of wise Scriptural child rearing of the wise, while fools despise such correction.