If I really want to sell my car, and I know it's in good shape, I would be more than happy to arrange an inspection. If the seller wants to avoid an inspection, it's probably because 1) the vehicle is in enough demand (or cheap enough) that it's not worth the hassle, or 2) something's wrong with the vehicle.

"Everything looks good, but I can't in good conscience buy this until a get a mechanic that I know to give it a once over. Can we arrange that?"

Don't buy anything from anyone that balks at this, regardless of reason.

Yes indeed. I got both my cars checked. The courteous thing to do is take it to a reputable mechanic (I took them to dealership service) coz if I am the seller; I would not like to take my car to a shady place even though I don't mind it being inspected.

Both times I've done this I just left my locked car at their place while I took their car to get checked out. It was no big deal. Set up an appointment with them and an appointment with the mechanic so everyone is ready to go!