I'm unsure if it's the cause, but it looks like clicking on my uncommon (and currently operating) drills removes the storage which causes the drill to stop operating until it's re-placed.
If I don't click on the storage though and only hover it, it seems fine.
EDIT: Having started a new save and cheated a bunch of different qualities in, it does appear to be related to the "quality" mod, as the internal storage is not flagged as being the same quality as the drill and remains "common".
EDIT 2: removing a common drill though with it's inventory intact does not yield the contents of the drill either (if I have 250 copper in it, I'll be given none when removing the drill).