One viewer challenged us with the task of taking a scratch out of a ceramic tile, without replacing the tile of course.
Simple buffers didn't do anything. Buffers with polishing compound did a little better. A soft rubber based grinder did the best, but left a rough appearance. A grinding stone went right through the colour glazing and exposed the clay below -- woops.
So a rubber based grinding wheel followed by a polishing buffing and great care might mask the scratch and not feel bad to bare feet, but even at that this is no longer a perfect new tile.