Would not recommend the diamond pads . . . you'll eat through a ton of them. We just cleaned 21 shelves from a salt firing . . . used carbide grinding stones, stiff putty knives, chisel with hand protector; so serious glaze runs so the angle grinder stayed at the workbench. For small glaze drops, a flat-headed screwdriver and hammer work. You just need to learn how to angle the knife/screwdriver/chisel so that you catch under the lip of the glaze bead. For bad glaze runs, grinder.
And, in all situations . . . wear your respirator, eye protection, and work gloves (no cheap cloth gloves); slivers of glaze can cause serious cuts. And never run your bare hand over the kiln shelf to see if there are any glaze spots that need removal.