Looked for a few minutes online and Missouri laws defer to the local municipalities/counties. Couldn't find anything specific for the city the golf course is in but I do know county wise there are kids all over this place buzzing around in ATV's, side by sides and golf carts. I mean, he shouldn't have been driving the cart. I let him do it a few times on the course and the 50-60ft straight drive over to the cart return theoretically wasn't all that dangerous....but I highly doubt he's the first or last 13 year old to drive a cart on that course or any other one. Not that it makes it 'right'.
And I think that's probably why this guy is digging in the way he is. He's a retired police chief who is most likely pretty confident where he stands on this. I spoke with a Sherriff buddy of mine who is a Sherriff of the County this took place in and he told me criminally, there's nothing that can be done to me if I just choose to pay half. Especially since there was no report created about the incident. But, obviously there's a chance the guy would take me to small claims court for the other $375. So really, this is about how likely do I think it is this guy would take me to court over $375?? I think a phone call and speaking to him would help me gauge that. It's also a case of how big a deal do I want to make this? Yes, $750 is a lot of money but at the same time....I have it and can just pay it and be done with all this. I'm more fixated on the principle of the thing that is there were and are two responsible parties that led to this accident, not just my son.
I've avoided weighing in on this because so much of this is location specific and fact specific. And with the proviso that this is not legal advice, and I'm not representing you, what strikes me is...
All this comes down to a matter of proof, and neither of you have a ton of it, EXCEPT for the video. Hopefully you saved the video. Does it have the driver agreeing to the damage assessment? If the driver isn't the owner, it's not going to be the be-all-and-end-all, but he is a representative of the business. If you DO get into a small claims situation, you have the VIDEO, you have the other driver's account that it was your son on the cart (probably not in your favor, though I haven't found anything specific on that; you should probably call the golf course and see what their rules are) and you have your account that the truck was speeding and I read somewhere something about alcohol. It depends on the judge, but in cases where it's a lot of "he-said, she-said", the judge will often rely on that which he/she KNOWS. There is video. There is the fact that neither party felt this was bad enough to call the police (though pro-tip: if the other side is asking for your information, you SHOULD consider asking for a cop to be present; Bart might disagree with me on that but once you have insurance companies involved, it's a whole 'other ball of wax.)
Do you have any other witnesses to the scene? You actually, while biased, are a witness (the driver is an actor in the situation as is your son). Anyone else that can corroborate your side of the story? Does the video show or indicate any impairment on behalf of the driver? Does the video implicate your son to any greater degree than you've indicated here?
Small claims in Missouri cost less than $50 to file, and they have five years to do it. I would stop with texts that say "make this right"; I know you were operating in good faith when you said it, I'm not being critical, and I don't think it hurts you too bad that you said it, but don't keep doubling down. I would say things like "accept my half of the responsibility" or something like that. Decide what you're willing to pay and document that. I wouldn't send this guy any more texts just yet, but when the bill comes back from the auto repair shop, I would pay your half - half seems fair - and have a response prepared, something to the effect of "In reflecting on this, it's clear that both parties had a hand in what went down on that day. The video taken at the scene reflects this. [assuming it actually does] I realize that to "make this right" from my perspective was to handle half of the responsibility. I disagree that this had to cost $750, but in the interest of settling this, I have paid my half of the responsibility, or $375 in anticipation that this will close the matter."
And leave it at that.
(And by the by, I'm assuming your son is safe and unhurt; that is the big thing and I hope it's true.)