Hi Jocke,
my watchmakers also uses such a tool. In his opinion flawless screws slots after a movement has been refurbished are a clear indication of a careful execution. A less thorough watchmaker will use one size of screwdriver for any screw and inevitable damage the screw slot. After a while a sharpening of the screwdriver becomes necessary like you have to do with a pencil.
So, just by looking at the screw slots the layman can get a first idea how carefully a movement had been treated in the past.
Kind regards and thanks for showing us a photograph of the sharpening machine!
anaesdoc