Sounds like it could also be a defective master cylinder. But plenty of things to check first.
the entire system's been redone
Just recently? By you, or a shop? If a shop, did they specialize in older vehicles, or are just a regular shop?
Stock single reservoir master, or dual/tandem master?
Do you know for sure if there are primary and secondary shoes, and if they were installed with the primary shoes in the front position at each brake?
They've also already been bled and adjusted...
This is the part that is concerning. If it bled well, and it still acts like it's either got air or a worn out piston seal, then it might be harder to track down.
Did you do the bleeding? Does it have a firm pedal with the first pump while sitting?
And how tight are the brakes adjusted? Do they scuff/rub the drum pretty good? To the point that you can feel a slight resistance when turning the tire with it up in the air?
At this point it would be good to know if there are primary shoes where they should be. This actually does help a lot with the pedal feel. Not having one does not necessarily mean you will have bad braking, and it normally would not give the symptoms you site. But it's still worth checking.
Good luck!
Paul