Here is a small diameter drive center I have made for my wood lathe, for driving small diameter pieces with a small and not too deep print in the wood.
It's made with a standard morse taper MT2 cone, sold for holding a drill chuck.
I had to drill at its end, and it's hardened steel. I managed to drill an 8 mm hole using a carbide bit made for drilling glass and tiles. I did that on the metal lathe, with lubricant.
Then I adapted a little toothed crown I found in one of those kits sold for fixing tap seat issues.
This is the idea, these crowns are ideal for driving the wood, and they are not too aggressive but hold it very well.
The center has a threaded hole, you can even change them as you wish, as there are several sizes.