Clout Nails
The Process you describe is called Clout Nailing. I grew up with this method and used it long before drywall screws and screw guns came on the market. The nails usually used are cut nails similar to carpet tacks. Because they have a
flat side, when they hit a hard surface after being driven
thru the wood, they
bend and curl up upon themselves thus locking the
nail into the lumber. Very effective way of fastening keystones and corner blocks in making a soft
flat.
Benefits of this method over using normal nails or screws is that nails won’t dig into the lumber or lie
flat when bent, drywall screws in that
thickness of material won’t have much holding
power and it’s difficult to have the screws
countersink into the lumber without them stripping out in the hole, unless unless you
countersink. You could also use longer screws and grind off the tips. Other option would be to use narrow
profile head drywall screws or use wood screws both of which have better thread patterns in the case of a wood screw for it's short length, or a narrow head easier to sink into the wood.
Clout nailing for the most part has been replaced by use of pneumatic staples which when used with wood glue will hold the joint together until the glue dries just fine. The glue attaching the keystones and corner blocks when they are run with grain crossing the joint is what holds together the joint. That is unless the joint gets wet or is flexed, than it’s probably the fasteners that are primarily holding it together until it fails.
You can use drywall screws, but it will take longer than using pneumatic screws and it takes much more work in replacing stripped screws or cutting off the tips that poke
thru. Drywall screws also have the chance of rusting a
bit easier with the glue and
sizing used in making up the soft
flat.