I did a little digging this morning, concentrating at first on Snap-on.
It looks the original kernel for the idea of the F-40 actually may have started in 1937. Here is a composite excerpt I made from Industrial Catalog 37P.
Note that the part number for the model with the 3/8-inch drive stud is GF-7, and it looks like it's being marketed as a special "Screw-Tite" tool for radio service, not as a Ferret accessory, although giving it the "F" (for Ferret) vs. "M" (for Midget) prefix is certainly revealing of where they ended up going in 1941.
I could not find any other documentation for "Screw-Tite". (If that's not their spin, pun definitely intended, on Walden's "Spintite", I'll eat my hat.) It doesn't look like the moniker lasted too long. Snap-on didn't TM it (a nail and screw manufacturer would in the 60's) and I can't find them using it in any other marketing (trade mags, ads, etc).
In 1947, in Industrial Catalog 47P, the only "Screw-Tite" listed is the GM-4 1/4-drive model.
The F-40 was not included in any sets in 1941 and the way it was introduced seems more like an accessory.
What's most interesting for me is the "Grip Handle" terminology.
Not postulating that there's anything more to it than coincidence, but Blackhawk always used that
exact term for its spinners, including the 3/8-drive one they introduced in 1941.
