I have that mandrel.
The 23/64 drill bit provided with the mandrel assumes that you will simply force the blank onto the mandrel stud, with the stud serving as the tap to thread a hole in the blank. Fine in theory, but if you are using a hard wood,, the theory breaks down. In addition, a 23/64 hole threaded for 3/8x16 doesn't leave a lot of thread depth when the time comes to attach the finished blank to the metal stopper base.
My preference is to drill with a 5/16" bit, and then use a 3/8x16tip tapered tap to thread the hole. After cutting the threads, drip a little thin CA into the threads, and allow it to cure thoroughly. This will reinforce the threads so that they are less likely to strip out when you mount the blank on the mandrel. After the clue cures (THOROUGHLY - you really have to be patient and just wait), run the tap in again to clean up any fuzzies on the threads.
And like Edgar, I prefer to put a recess in the bottom of the blank so that the stopper seam between the stopper blank and the bottom of the blank will be hidden by the edge of the recess. I have used forstner bits, but find that its easier to cut a small recess with a gouge or scraper when facing off the bottom. This step makes the final product a bit more 'finished'