The setup/assembly as pictured looks about right to me, Matt, complete with the thrust washer at the rear, behind the trailing caliper.
The basic problem with this sort of stamped-steel sidepull is the thin arms in the same plane as braking forces and they aren't very rigid. They clamp "okay" but judder under braking as they really can't resist the forward movement of the pads against the rim very well.
I'd say lube, adjust out any excess play, lock in the adjustment and call it done. Fitting new/uprated brake shoes can actually make braking worse with thin-armed, long-reach stamped sidepulls like these. I know...I've got a set on my 1970 Motobecane Astra U-frame Folder.
Best, Dan.