As well they should.
I think they should.
I have bought 2 cars from private sellers. I was interested in another car that I did not buy based on the work that my mechanic said needed to be done. Car was selling for $200 and it would need about $2,000 in mechanical work plus if I chose to do so, some body work. Nope.
First car was a 1970 Ford Maverick. I had that thing in the shop about every two weeks. Horrible car. Main problem was that it had disc brakes that were realized to be, several years after I bought it, inherently bad.
Other car was a 1974 Dodge Dart. Although it checked out as being good, it threw a rod 2 weeks after I bought it. Mechanic said it was a fluke. I put in a new engine, my dad and I had done some body work prior and had it painted. We did the trim ourselves. And I had no real issues or problems with it for several years after.
Next car was actually purchased from my mechanic. No problems with it whatever but normal wear and tear. I sold it to my BIL who is a mechanic and used car dealer for about what I paid for it. That's how good it retained its value. I also bought two vans from him. One had an issue right away. It was some sensor. Not a real problem but it was making the brake light come on and we were in the middle of a cross country move so that was worrisome.
I have a brand new car now.
I would never do this. My friend did buy a car at an auction and you can't really take it to a mechanic when you do that. It worked out well for her but that isn't always the case.
I have been to many car auctions with my BIL but those are for dealers and not the same as for ordinary citizens. But I know enough what to look for at an auction. Stuff like worn belts, dirty engine, rust, dripping fluids. Also if the car is allowed to be started to look for odd noises and if you can put your foot on the brake pedal to check for mushy brakes. Keep in mind that you can't always start it up.
That being said... I still would not personally buy a car like that unless I could do some mechanical stuff myself, which I can't. Things were a lot different with the older cars that didn't have electronics in them. Back then I did help my dad do some repairs. I wouldn't try it now.