The spindle is no great problem as long as any centre fitted has no run out. It will revolve with the part it is centred on or could/would have a drawbar to tighten the morse taper drive otherwise. Tha tailstock is a slightly different problem in that it needs to resist turning and be in line with the spindle centre.
Wet and dry have a thickness, so any morse taper will not clean out the full depth but must be good for most of the taper. Remember, too, that these tapers may or may not be hardened. The problems start when swarf has been trapped previously or the surface is badly galled.
So my advice would be to start with the least aggressive methods and progress from there. Testing for contact at each stage is important, to know where the high spots are. If resorting to a reamer, then removal of the absolute minimum of material should be aimed for, I would think.