This post reminded me we used to have two Ward capstans in our workshop, mainly used for making studs and bolts. And I can't remember greasing them. I used to squirt the ways, rotating and sliding surfaces with machine oil.
I would think grease is too sticky for a capstan, doesn't flow where you want it to, and would slow down the operation, speed is essential when working a capstan. Oil will also flow better and not turn into grinding paste.
Hope this helps
Ian