I think the "dump a car when repairs equal its value" rule of thumb only really applies when someone has the means to buy a new, or significantly newer/better vehicle. If it means dumping one vehicle worth maybe a couple grand for another vehicle worth about the same amount, it doesn't necessarily work out. You end up buying a new set of problems.
Personally, I would put more than my truck's actual cash value into repairing it if need be. For example, a transmission would realistically total it out. The body has a lot of damage from a wreck four years ago that would be expensive to repair, it has over 150K miles on it, and it's 12 years old. In perfect cosmetic condition, a transmission would make sense from a straight cash value perspective. In it's current cosmetic condition, most people would send it to the junkyard if it needed a trans. For me though, I know the truck from front to back. I have service records dating back to brand new. I know literally everything that has been put into it and that has happened to it from 15 miles on the odo (parents owned it to 49K, then sold it to me). I can't possibly get that with any other used truck, and I have high standards for maintenance that most previous owners won't meet. Most used trucks would require work to get them up to my level of mechanical maintenance and care, so I factor that in to the equation. Basically, if the trans goes, at least I know where everything else stands and can get it back to that point. If I buy a used truck, the trans could go the next day and I'd still have other things to deal with. Despite how ugly it is, it would probably take the engine and trans going out at the same time for me to junk it, which is unlikely. A wreck that finally finishes it off would probably be the only thing that would send it to the crusher since rust isn't happening. Plus, I actually feel more attached to the truck after wrecking it. Weird way to bond with a vehicle, but I didn't have so much as a cut from the wreck until I got out of the truck and made my way up the ravine it landed in.
This would not be the case if it were badly rusted. I did throw in the towel on my old Explorer, which ran and drove great, because rust was eating it alive and there was no stopping it. The Ranger looks bad due to the crash damage from years ago, but the body and frame are safe and sound. My Explorer was disintegrating before my eyes, and I had to accept the hard reality that I could not save it.
Rust is really a major factor in such a decision because it is the only thing aside from maybe flood damage that can just totally ruin a vehicle and cause multiple major component failures. My sister has a 2006 Mazda 3 she bought new, and she lives in VT. It may be getting junked this year due to rust unless someone else feels inclined to save it. The car has been well maintained, but will not pass its next inspection, and to her is not worth the thousands in repairs needed to keep it on the road. A new car makes more sense to her.
Now for people who are lackluster with maintenance and just run something into the ground, maybe swapping for another low value vehicle could work out, but they will be back in the same situation quickly.
Basically the rule of thumb is whenever you have had enough.
FWIW, Pull-A-Part has started hitting up the auctions, and they play hardball. I talked to one used car dealer who told me, "you used to be able to get runners for $200-$300 all day, now because of Pull-A-Part every car in Birmingham is $500." Look at the writing on the windshields the next time you are in the yard, it can tell you a lot. I have seen a lot of cars there that were sold as runners and probably could have been made decent cars for a grand or less in repairs. I saw a 2002 or 2003 Infiniti QX4 the last time I was there that actually said "runs/drives good, cold A/C" on the windshield and had a fairly clean interior. It had damage to the doors on the passenger side, but the actual body was straight. Two new used doors painted to match would have made it a nice SUV, but it was in Pull-A-Part. A lot of people are driving cars way worse than that QX4. The salvage business is a strange animal.