kxxr & others interested,
After an exhaustive catalog search I finally found this tool. It is not a screwdriver, or a burnisher, or a steel. Are you ready for this?
It is a no. 6497 grinding wheel dresser. It featured a diamond tip (don't know if yours still has the diamond) and was sold from 1944 - 1954. The one you have with the narrow fluted handle with a ball on the end was sold from 1952-1954, and then this type of dresser was discontinued altogather.
Who would have guessed?
EDIT: A bit of trivia-- Why did this wheel dresser disappear? We'll probably never know for sure, but I note that the diamond tipped dresser cost $4.25 while the more conventional two-wheel dresser cost $0.95 in 1954. So it could be a case where it was just too expensive, and consequently, few were sold. The conventional and less expensive two-wheel dresser design, on the other hand, is still being sold today.