If you have solid fluid at the wheel cylinders when you bleed the system, then my thought is that the rear brake shoes need further adjusting, or that the car should be driven a bit. In the old days shoes were arced to produce a radius on them that matched the drum inside diameter. That's rarely done today, so the shoes may not be a good fit yet. That can account for the springiness you detect.
Be sure they are adjusted out tight, stamp the pedal a few times, and back the adjuster off to a light drag. Drive the car if the pedal is at a safe height. You may find the brakes wear in fairly quickly.