Right. People tell me "but how can you trust a perfect stranger?" To which I reply most people selling cars are honest. Most car salesmen are dishonest. I trust private sellers more than I trust car salesmen.
A couple of times, I've been fine with buying the car first, but with a clause in a sales contract saying that I have the right to return the car within 3 days for a full refund (basically replicating the effects of my state's lemon law. The lemon law only applies to new car purchases).
But the mechanic's check has never actually revealed any issues that the seller hadn't already disclosed to me. People tend to be pretty honest about those sorts of things.
"I'm looking to buy a used car and wanted somewhere to give it a once-over? Do you do that? If so, how much and do you have availability on (day)?"