what are the benefits of going through a dealer as opposed to a private owner?
Depends. The dealership's mark-up is always going to be higher than a private owner. So, you can try to haggle lower with a private owner. But as someone has already said, they are going to sell it to you as-is. But in reality, so do used car dealerships. You have an option to ask them to fix something. They don't have to and sometimes, you are better off haggling for a lower price rather them have them fix it. What you did is buy whatever part and fix it yourself.
What "as-is" means in legal terms is you are buying it in the state that it is in now. If something breaks after the sale or is already broken now, you bought it in that state
whether or not you know how to test it for functionality. This is why you need to test drive the car too.
This is why you need to know what to look for. Really, you need to go with someone who has been around and knows what to look for. There are too many things.
Simple stuff is check all the door locks, window switches, power mirror switches, sliding seat lever, windshield wipers, windshield washers, trunk release handle, gas tank cover release (if it has one), hazard light, turn signals, all gears, emergency brake, clock, radio, glove compartment door, headlights, high beam switch, fog lights if it has them, rear window wiper if it has one, heater/air conditioner fan. (don't feel like a goof ball when you are being extra thorough. You are buying a used car as-is. It's going to suck when you find out one of the switches don't work after the fact. A simple press of a switch would have uncovered that problem. Hop in the back seats and operate everything as if you were a passenger. Have the seller turn on the turn signals and brake pedal while you are standing behind the car so you can see them turn on properly.