"What drill bit is it that drills a pilot hole and sinks the screw so screw heads are sunken and not flush or showing?"
This looks like a project that means a lot to you and probably others. Although it is a lot more work, the results will be more esthetically pleasing than exposed screw heads. Long ago I started using a 3/8 flat bottom drill (forstner or similar) and then a plug maker. I take a piece of wood that matches the wood I am using and then cut plugs from it to use to fill the 3/8 holes.
A little glue, tap them in with a small hammer, and sand.
I can tell you from some painful experiences, the best way to cut the plugs is with a drill press.
