We got ripped similarly last year. We'd been in an accident and been 3 days without our now-totaled van. We have too many kids to fit us all in anything but a van. We finally lined up 3 test drives in a day, with a babysitter for the kids, and no one here wants to let you get an inspection either. When the price tag is under $3000, though, we expect
some problems. The van seemed wonderful, drove well, seemed to have no problems at all. The ad on CL had said "NEEDS NOTHING!" and that was exactly what we needed. My father was a mechanic, so I know the basics to look/feel/listen for. The guy we met was a "friend of the seller". He worked at a dealership and reiterated "to my knowledge it needs absolutely nothing."
The guy took a little less than asking, and offered to write in $100 on the title so we could get a break on the taxes. He seemed great, and we gladly took the offer. Next day the check engine light came on.

Turned out the transmission was shot, and the guy had turned off the light. What it looks like is the guy was a salesman at the dealership, the dealership had taken the thing on trade and pawned it off on us. Luckily, a $1700 rebuilt tranny later we have a great van still kicking on 160,000+ miles. Were betting on having to buy another van next year, and will do so before we
need it so that we can afford to turn down someone who won't allow an inspection.