Just to conclude the story. The steady rest eventually arrived after a week and a bit and works great (but if I ever make one of my own, i'd definitely give it 4 fingers so it's less of a fiddle to adjust), but even after that there was still about 0.60mm wobble on the end of the shaft. This was caused by the motor's own spindle that I couldn't grind better than 0.01mm using my plywood setup and dremel knockoff.
So I just scraped the shoulder of the spindle till I got 0.20mm wobble, which reduced vibrations to an amount I could live with.
With that the spindle sander is now working. I just need to pretty it up and figure out the dust extraction because whoa nelly.
Thanks for all your help!