I will second the motion to shop by mileage within the broader category rather than by make/model.
1) If you are shopping for a $3000 car, you could have a Fit with 200k miles or, say, a Versa/Spark with 100k miles. Which one do you think will face a major drivetrain failure first? The truth is, it's a gamble! On the one hand, a top-quality car that's completely worn the fuck out, or a car with a slightly worse reputation but far less wear. The market is efficient because people at this price range are thinking very hard about the financial implications of their decision, unlike new car buyers. At a given level of size/status symbol, you get a given level of reliability. Example: My SIL found a $2500 Corolla that was about 13 years old with only 90k miles. Top brand. Low miles. Seems logical. Transmission was toast three months later.
2) Mechanical problems are distributed across a car's lifespan. Design and manufacturing factors create a risk, but not a certainty, of an important part failing prematurely. Also, there are tens of thousands of such parts in any car. The cars we call unreliable have a handful of parts with a high risk of failure. However, the parts either fail or they don't. Statistically speaking, some subset of "low quality" cars will provide their owners decades and hundreds of thousands of miles of reliable service because the entire handful of vulnerable parts happen not to fail in that particular car. Also, some subset of "high quality" cars will be disappointments because of one improbable but consequential failure. I've seen Aveos running 200k miles and I've seen nearly new Toyota's with blown head gaskets. So reliability is driven partially by chance, but wear and tear are easily measured certainties.
3) A Honda Fit with a blown engine is worth more than a Chevy Aveo with a blown engine. This is because when a Chevy Aveo gets into a minor fender bender or bends a valve, it typically goes straight to the scrap yard (increasing supply of parts with no increase in demand). Why spend $2k fixing a car that will be worth $2k after your labors? When a Honda Fit gets into a fender bender or suffers mechanical failure, it's higher fixed-up value justifies the repair expense (increasing demand of parts with no increase in supply). It makes sense to spend $3k fixing a Fit that will be worth $5k after the repairs. So part of what you are paying for when you buy a Fit is the high price the salvage yard could get for your fenders, doors, and drivetrain. This is also why you see more Fits with 200k+ miles than Aveos; people keep fixing them instead of scrapping them. So to buy the fit is to essentially lock away money that you'll partially get back at resale time or when insurance totals it. On the other hand, you could buy a younger, lower mileage car for the same money, and arguably get more reliability/longevity for your money.
We have a 2012 Fit that we bought new for $16k. It is now worth about $7500. My main complaint is that the seats are so uncomfortable we don't want to use it for long trips. I mean sadistic uncomfortable. My DW got a horrible back spasm on a 6 hour journey. That's a shame because this is exactly the type of car you'd want to use for long journeys. In all our analysis of costs, reliability, depreciation, new vs. used, and how to avoid dealer add-ons, we failed to think about how uncomfortable the seats are! Now I think a lot more multi-dimensionally about cars than I used to.
If I was shopping in the $3k range, I would look for lower-mileage, later-model cars that have salvage titles, body damage, rust, or bad paint. Those cosmetic/fear factors lower the price but don't generally affect the drivetrain. Also, DO NOT STEP FOOT on any car lot unless the cars are sitting on dirt or gravel. Negotiate hard if you have cash, and remember that at this price range any dealer is making a 70-100% markup - i.e. they paid $1600 at the auction for the car they're offering for $3500, so don't be shy about offering them $3k cash.