I first learned using a barrel trimmer. A hand drill or drill press worked OK. Downside was having to buy or make the correct sized sleeve to fit inside the brass tube.
I had better results by first turning the blank round to about 3/4". I then could chuck the blank on my lathe and drill a centered hole for the brass tube using a Jacobs chuck on the tailstock. For squaring the blank ends after the tube was glued-in, I would re-chuck the blank in my lathe and now use the barrel trimmer mounted in the Jacobs chuck on the tailstock. Since the barrel trimmer is held centered on the brass tube, there is no need for a sleeve. I still had a problem with the finish on the milled end. Since we are dealing with end grain, it would often be less than smooth. I also experienced some chipping with laminated blanks (Dymondwood).
I ultimately went to using a disc sander with its support table and miter gauge to keep the blank square to the sanding disc. I find this to give me a better finished surface and is faster than using the barrel trimmer. I no longer round my blanks first, but am able to quickly sand off the sharp edges on the sander (for my Tru-stone and acrylic blanks). My shop vac provides excellent dust control. I find my disc sander to be one of my best 50$ investments from Harbor Freight.