Hi J27 - I've experience with polishing dies for aluminium tube drawing and automotive press tools. The smoother the finish the better and as Martec says the final passes need to be made in the direction of draw. We finished tools with diamond paste, I forget the grit was decades ago. So down through the paper grits to 2000 plus then move to SiC grit then to diamond. When trialing tools they did not have final finishes but it is noticeable how the diamond finish produces better surface finish to the prior grits. I expect it comes down to the lubrication film thickness. If the surface roughness is higher then the film thickness you have surface contact. Once the roughness is less then the film thickness then you have hydrodynamic lub so no metal to metal contact. The finish is a mirror finish to achieve this.

The machinist use a tool like a jigsaw that held mop or stick and he set the tube dies in the lathe and it rotated slowly. That's fine for a round tool. The car parts are odd shapes so had to be polished by hand. takes time but patience gets you there. The use of surface burnishing is more about setting the surface into compression then the finish itself. If you have a tool that needs to make millions of parts they will fatigue and the surface will flake off at some point ruining the tool. Look up hertzian stresses. So by working the surface and creating a compression zone the tool lasts a lot longer. Same principle as peening welds. I expect they are still polished after setting...Peter