What are you cutting with?

I prefer to ctr drill with a very small Ctr Drill and use tailstock support so the shaft can be turned in one go. I use a DCGT insert with 0.2mm tip radius and that allows fine cuts to be taken without deflecting the work. If that does not work then polish to finish with fine emery backed by a block.

You can leave extra length on the stem if needed for clearance and to take the hole then cut off and face to length.

I've only silver soldered larger valves but doing it all at one setting from soliid metal gets everything concentric

There are other very sharp inserts with a 0.1 or 0.08mm radius that are very good for taking most of the cut in 1 pass. I will often leave 3mm on diameter for the last pass and use the 1st pass to take the outer skin off the stock and use it as a gauge for the diameter it is cutting at. Making a Vee block polishing tool , works a bit like an external hone. Allows the polishing while keeping the part round. Often patience is a good thing and not too high a rpm. Keeping the speed down, will keep the heat lower and less distortion of the part.