If I leave my strings on too long, they start to exhibit strange overtones. Still, I can't think of a connection between that, and intonation on just 1 string being such that, improving it for one area of the fretboard would make it worse in another area.
But nevertheless, I would check the tuning at various frets (even compare 2 different tuners), write down the values, then put on new strings and repeat. I always try to eliminate any obvious variables first. I mean, if you had a cable that occasionally made some noise, and you were trying to track down a tone issue, even though it may seem like the cable isn't affecting the specific problem, it just makes good sense to get a new one as a first step.
That's my diplomatic way of saying, "Dude! You have a problem with strings being in tune, yet they're old? Well duh, start by changing them."
Also, how much relief is in your neck? The intonation adjustment accounts for how a string's amount of tension increases slightly when pressed to the board. Maybe, and this is just spitballing, a neck with a large amount of relief would affect intonation along the length of the board, since the distance from the string to the board would not be as linear an increase, the further up the neck you go, but instead be a certain amount at the ends of the board, and yet even larger amount near the middle.
And I have checked intonation at various frets too, and never found it to be exactly correct everywhere. Even though your example seems to be especially weird.
With the amount of pitch change, the relief you can see in that vid would have to be like 50 times what it is to go 70 cents off at the 12th fret.... He said"I dont know" when asked about the string's age, so Im betting that means sometime during the obama presidency
could be a lot of things. first, play the guitar in the position you are gonna hold it in. second, change the strings to new ones. third, it could also be just the way you are playing. if you press too hard or too soft, that could also be part of the issue. the video doesn't really pinpoint it.