Private party sales of cars in the USA use cash, yes. A fairly safe way to do this for a car is to do the sale in the lobby of the seller’s bank or credit union. In specific detail, the buyer arrives and inspects the car to their satisfaction, they fill in most of the vehicle title transfer form but withhold the signature, then they go to the teller where the buyer produces the cash, the teller counts the funds and deposits it to the seller’s account, and finally the buyer and seller sign the title transfer.

In this way, the buyer’s risk is minimized (they’re meeting in a quasi public place with cameras, so gunpoint robbery would be unlikely) and the seller’s risk is minimized (they don’t even have to handle or count the cash, so a bad buyer can’t even rob them after the sale).

Not sure if this would be applicable to OP’s situation, but it might be adaptable if the seller’s bank is nearby and each party makes their own way to the bank to complete the sale, after inspecting the object.

Note: if instead of cash, the buyer pays with an official or cashier’s check, then this whole procedure is still valid because the seller doesn’t have to deal with the risk of a bad check: their bank has procedures to verify on-the-spot the funds on an official or cashier’s check issued by another institution. For a personal check though, such checks have a TOCTTOU problem, so sellers should not accept such checks for the sale.