You are correct, the drive center which will most likely be the spur drive that came with your lathe will leave a tiny hole and some other marring on it's end, and the live center will leave a tiny hole, and perhapse a ring shaped dent on the other end.
There are a methods of dealing with this issue that do not involve extra equipment like chucks.
Assuming a typical bottle stopper kit, you will need a hole for the threaded insert which will be drilled before turning. as long as the insert piece will cover any marring at that end of the project, you don't need to worry about that unless you get bad enough marring to make a bad looking joing between the insert and the turning.
One approach is you make the starting blank a bit long, and part off the excess material. With care, and light cuts with a gouge or skew you can turn down the end opposite the hole to a very small almost thread like amount of wood supporting the blank that you can then cut off and sand smooth.
You can make a morse taper drive. Get a hanger bolt that matches the threads of your insert. Drill a hole in the end grain of a piece of closed grain hardwood like maple that matches the wood screw end of the hanger bolt, Turn a taper on the oppposite end that matches the taper on your spur drive. Screw in the hanger bolt. Insert this taper in the head and use it to drive the drilled blank.
If you have a Jacobs chuck for drilling on the lathe, you can use a regular bolt with the head cut off in the Jacobs chuck as a drive center.
With both of these screw drive approaches, you would use the live center for support until the very end when cleaning up the opposite end.