And now, someone is going to go and look all these up, and say, "oh, no, you see the literal translation rod means stick less than a quarter inch, and probably actually refers to pretzel rods of some sort." Don't. Don't fight it
Sorry, I'm the rebellious argumentative type and have to.

I understand this is what you believe and that's okay we can disagree. I don't want this to get all inflammatory, but just want to share some thoughts on this. Not to take anything away from the disgusting show of human filth shown in the original video.
Instead of the pretzel rod you're suggesting it is a
scepter, literally that is what the word means in Hebrew. Scepter, you know the golden wand kings wave around to symbolize their authority. What's going on here is a metaphor, something extremely common in ancient poetry of Proverbs.
Folly is close to the heart of a child, but the rod of discipline will drive it far from him.
(Proverbs 22:15 NAB)
This does not mean beating or physical violence. It does however mean that parents have the responsibility to discipline their children. This means that if a child initiates force against someone else, the parent needs to communicate to the child how their actions have consequences, and then have the child makes up for their bad behavior in a controlled and helpful manner, such as through restitution. This makes sure the child is prepared for the real world where you don't get off so easy for punching your boss, as you do on the play ground.
Blessings are for the head of the just, but a rod for the back of the fool.
(Proverbs 10:6 NAB)
The Hebrew word here translated as "rod for the back" is literally violence, the NAB is taking some serious liberties here; most translations read something like, "violence overwhelms the..." But anyways the language here is not commanding us to dole out the punishment, but rather that the violence is a result of the person's own actions, in other words the fool has brought this on himself.
Withhold not chastisement from a boy; if you beat him with a rod he will not die. Beat him with the rod, and you will save him from the nether world.
(Proverbs 23:13-14 NAB)
He that spareth his rod hateth his son: but he that loveth him chasteneth him betimes.
Proverbs 13:24
Same deal as before with the rod being metaphorical and not violent. Chasten in both places is a translation of the Hebrew for discipline.
"Happy those who seize your children and smash them against a rock."
Psalms 137:9 NAB
First some background, this Psalm was written after the Israelite nation had been taken over by Babylon. This song/prayer is from an Israelite angry that God had allowed them to be treated this way, asking God to deliver them. If we read the verses immediately preceding this quote:
"Daughter Babylon, doomed to destruction,
happy is the one who repays you
according to what you have done to us."
The Bible isn't saying lets go throw babies off some cliff, or even throw some Babylonian babies off of a cliff. It is one man crying out to God for retribution (similar to what we find throughout the book of Job), not God commanding retribution. Only God had a different plan that didn't involve murdering babies, but that's another discussion.
Chastise your son, for in this there is hope(Proverbs 19:18-19 NAB)
From the ESV, "Discipline your son, for there is hope; do not set your heart on putting him to death."
The argument that religion is something we can just ignore is not okay. I agree with Stef here. The fact that that father is a backward, childbeating control freak has to do with the bible. I guarantee it. He lives in the bible belt, and I'll bet you anything he's [claims to be] a christian and goes to church.
I agree almost completely with what you say here, with that one small change; anyone who actually lives by the golden rule could not treat a child like that. The fact that the man was (is?) a Judge drives the point home. Organized religion where people just sit back and accept whatever the Preacher spews out at them is just as dangerous as the people who listen to the despots.
One needs to read and look at things from all sides. That was the way I was raised: to look at everything skeptically and draw my own conclusions. Luke even commend this in Acts where he speaks of the Bereans who didn't just accept what Paul taught them, but checked for themselves to see if it was true. Through healthy doses of Marx, Plato, and Rothbard I eventually found anarchism. In the same way through Rand, Nietzsche, Lewis, The Koran, and The Bible, I made the decision to be a Christian, it is the only view that makes sense to me. It may not to you, and thats okay, I'm not going to try and force you to do anything.
Like Stef said the antithesis to liberty isn't government, it is the initiation of force. In the same way the antithesis of liberty isn't religion, even though religion can be (and certainly is in many cases) involved in the initiation of force, it doesn't have to be. "Those who live by the sword, die by the sword."
Sorry for the long rant, hopefully it didn't come off as too argumentative, I just wanted to share my thoughts on the subject. I'll leave you with this:
Ephesians 6.4 "And the fathers! provoke not your children, but nourish them in the instruction and admonition of the Lord."