Did you maybe reverse "neck" and "body" in your description? Because the screws definitely need to thread into the neck!
In terms of the body, ideally the screw would make contact with the hole but the threads should not engage. In fact, any time you're screwing two pieces of wood together, it's typically ideal to have the threads only engage in the "far" piece of wood - threads engaging in both can lead to the screw wedging the two pieces versus drawing them tight.
Some wood screws have a non-threaded portion of the shaft under the head that's the same diameter as the large diameter of the thread to facilitate good contact with the hole, but in a Fender design with a bolt plate on the back that's arguably not as important.