Can't you just buy from a second hand car dealer? They usually give you more guarantees than private sales (for a slight premium). That's how most people in the UK buy cars.
You can, but it's usually a larger difference. A car that might be negotiated down to $5k or $6k on the private market, will probably be at $9k-$10k on the dealer lot. Plus they'll tack on document fees, and so on. In the end, you might be looking at paying double what you would otherwise for cars in similar shape and mileage.
That does also basically erase any connection to the last owner - so you can't try to infer if they took care of the car or not. On top of that, afaik many second-hand dealers buy cars at auction and then often only do basic (bandaid-type?) fixes to get the car ready for sale.