Ive had multiple cars I bought for <$1000 and also used to finance a new car every 2 years...DUMB DUMB DUMB cause I didn't have the income to take that kind of loss on new cars every 2 years!
Now I buy cars ONLY in cash and only at a BIG BIG BARGIAN, keep them till they are no longer worth keeping. Have had my 02 Bonneville for 2yrs, and my 89 Formula for over 8 years now. I will probably keep the Formula for ever and like I said the Bonne will go when the repairs outweigh the value.
I think you are looking at two extremes though. Managing debt is okay. Having unmanageable debt is not!
Some people have just accepted a car payment as a permanent part of their budget. And that's okay too, if that's how they want to do it. I'm tempted to do the math and see which way you actually 'win'. Whether keeping a car a few years after it's been paid off and trading it in with lesser value, or trading your car in before it's paid off (but when it's still 'young' enough to be worth something), keeping yourself in perpetual payment mode, BUT having less of a purchase price.
I know some people who have a seperate savings account and put a 'payment' in that account each month. They use it for repairs to their car and, eventually, to buy a new one outright.
As far as repairs outweighing the value, I think that's relative. If your car is an old, but reliable car and needs $2,000 in repairs (like a new transmission) but is in otherwise great shape; yet is only worth $2,000, I think you can still come out ahead. After all the only car you can get for $2,000 is one as old and high mileage as the one you are wanting to fix!