I think that's probably right. The one Widget put in the Museum is also unpainted veneer, but may have been painted originally, of course.
It's in the Museum: Shares
Yes. See Morris Shefras
Years ago I bought a couple of these late, unrestored Shefras allwins which were in ply cabinets with nice original oak veneers, so they did make them (one was a Payola). I assume they opted for paint as a further money saving measure at some point.