So two solutions have been suggested - epoxy putty to pad out the oversized hole and glue the insert into the blank, and gluing in a dowel that can be redrilled the proper diameter to receive the threaded insert.
There is a third approach that might be even better.
The basic challenge in making wooden stoppers is that they are typically spindle turnings where the fibers in the timber are lengthwise, parallel to the bedways of the lathe when the blank is turned. As a result, a hole in the end of the blank that is threaded using an ordinary tap will have weak threads that can shear out, either during tapping, or later as the blank is being turned. The conventional way to minimize the risk of this happening is to strengthen the threads by saturating them in thin CA.
Using a threaded insert is another solution. Threaded inserts come in two varieties - some have a coarse thread on the outside that are intended to cut into and grip the fibers of the wood., while others have a knurled outer diameter, and ideally are glued into the timber. In either case, however, they work better in face grain and are difficult to install in end-grain. My experience with them is limited, but in every end grain application, I've ended up having to use an oversize hole, and fill the gap between the insert and the hole with epoxy.
The third option is to make a wooden threaded insert from face-grain timber. Because wooden inserts are made from scrap, this solution doesn't cost anything (commercial metal inserts increase the material cost going into a stopper which has to be offset by a higher selling price if you are making them for sale).
The difference between a plug and a dowel is the direction of the grain - the grain in a dowel runs lengthwise, while the grain in a plug is parallel to the face of the plug. In furniture restoration, when it is necessary to repair a threaded hole where the threads have stripped out, the preferred solution is to install a plug rather than a dowel because new threads in a plug will be stronger than new threads in a dowel. A wooden threaded insert is nothing more than a face-grain plug that is glued into a hole in the end of the blank that is threaded to receive the mounting stud on the mandrel and later the stopper itself.
The way that I make these is to drill a hole in the face of a scrap of wood - I prefer timber with a strong grain like oak or ash. I thread the hole using an ordinary machinist's tap, and them screw the scrap of wood onto a stopper mandrel to turn the outside to a diameter to match a hole that I drill in the bottom of the stopper blank. Specifically, I drill a 5/16" hole that I thread using a 3/8"x16tpi tap, and then turn it down to make a 5/8" plug. Because the threads in the plug are face grain they are much stronger than threads in end grain; I never bother reinforcing them with CA. The glue joint between the stopper blank and the plug is a combination of side grain and end grain, but is more than strong enough for this application.
