.. a screwdriver bit in an old brace ... but a quick search reveals no source. Were these items once manufactured?

Yes -- perhaps an old tool dealer might have some.

Are they still available?

I don't know.

... with the old square section tapered shank for locking into the chuck of the brace?

If worse comes to worse, You could take an old auger bit, cut off the
drill section, and fashion the remaining shank into a screwdriver of
the size and shape you need.

Joel Jacobson