The last car I sold privately was to a genuine car dealer who travelled from Surrey to Devon on a train having given me a host of evidential details and he was happy to pay me cash and the take that cash to a bank in a nearby town and pay it directly over the counter into my account. Lovely bloke, genuine sale, no problems.
I started selling my (own personal) cars and motor bikes to private buyers in 1968. All without trouble - but that was a long time ago.
AND, YES, THERE'S A BUT ......
Without wishing to be the eternal pessimist ..... the private sale before that one was different.
I privately advertised, in the Essex/London border area, my 12 month old, one owner, very low mileage Toyota Yaris diesel ( it would have been something like a 2002 model ). I believe it was in a local paper that I placed the advert.
Somebody rang and arranged to come to the Underground station on the Central Line near to my, then, home. I offered to collect him from the station and duly did so. He got in the car, we drove along a main road through Epping Forest, he asked me to pull over into a pub cat park so he could look around the car.
And, in short, without going into detail, he carjacked me, and I watched my car disappear up the road heading back towards East London leaving me laying in the road with assorted injuries. The car was never seen again.
Clearly it was my own fault but after having sold loads of cars and a couple of bikes in the years previous I wasn't sufficiently wary or prepared.
So if the OP is able to take anything from that, then so much the better.