Hello Carl,
I too have cleaned up the morse tapers on tailstock barrels, but I adopted a much gentler way of going about it
I gripped the tang end of the correct size reamer in the bench vice in one instance and pushed the barrel onto it while rotating it by hand. You can 'feel' the cutting that way. I haven't measured the result as the lathe is being refurbished but taper shank drills fit with a nice 'kerthunk' noise and grip properly
In the second case, a 3 MT version, the tailstock barrel [18 inches long and heavy] was gripped via a cloth in the vice and the reamer turned by hand in a big tap holder, using the tang gripped between the jaws of the tap holder. Later measurement on a precision bar at 5 inches from the mouth of the barrel gave a run out of under 0.001 inches and again there was the satisfying 'kerthunk' noise on fitting drills
Regards
Brian