KNeilson tt looks like your "Archimedes drills" have collet holders, or maybe one has drill chuck. I am pretty sure that collets were used way before our time.
I have a small brass lathe that not only uses a system similar to collets, but kind of like the "collet" on an Xacto knife. The lathe is marked 1747, I bought it at a flea market style market in Spain in about 1963.
Back to the drill, we lived in Spain for 3 years, during that time I remember seeing a lot of woodworking machines that were older than anything I have ever seen here in the States. One I remember was a drill press that was hand operated, not like a blacksmith drill press, but it looked like a brace and bit mounted on a wall so that it would slide down as the drill went deeper. So that the drill was always at 90 degrees to the object being drilled. I don't remember the types of drill bits, but the homemade drill press has always stuck in my mind. In my mind the hole would have been drilled with some sort of jig, like a drill press so that the drill bit was always perpendicular to the piece of wood.
Hand operated and period correct? I don't know but it seems possible.
Looking at the Archimedes drills that KNeilson posted, I think there could be a jig built to hold it so that it stays perpendicular to the stock. That in itself would help prevent bit breakage, and the spoon bit would help insure a straight through hole.