On the system depicted in the above diagram, if the automatic adjusting lever, latch, and spring were removed there is no way to mechanically adjust for the shoe wear. With a lot of adjusters with the star wheel/screw strut or spreader, you could take out the auto adjuster mechanism and just periodically readjust the rear brakes. According the to above diagram there is no screw spreader to be adjusted if the auto parts are removed.
BTW, my experience with Corollas is that they have a star wheel adjustable strut with an automatic adjusting pawl that is activated by clearance taken up by the parking brake being applied. You have to use the parking brake to take up the rear brake shoe clearance. So if the parking brake has not been used in a while or the auto linkages have been removed, you are back to manually adjusting the brakes.
Hope this helps.
Most likely the rear shoe linings are worn, therefore when the brakes are applied the shoes have to travel a farther distance to contact the drums. That is why the brake pedal needs to travel farther to get the shoes to contact the drums. I bet if you gently pump the brakes when stopping the pedal rises and does not fall.
The shops only did half of their homework on this. They did not properly diagnose the low pedal. A bad m/cylinder would have a pedal the slowly falls when the brakes are gently held down.
This is no fault of yours but it is “brakes 101” when the shop has the car to look at.
This is not a front disc brake problem since those by design never need adjusting.
A zebra hoof beat to check would be a sticking front caliper. I’ve got a Geo Prizm (same thing as a Corolla) that I chased a soft pedal for quite a long time. Bleeding, adjusting did no good. Then, once when I had the fronts off for inspection, I noticed that one side had the inner pad completely worn, while the outer was nearly new. Turned out that the caliper was not sliding like it was supposed to, and all of the braking force was going only through the inner pad (no actual clamping.) Since the pad wore out much faster in this condition, I had to pump up the brakes to get hard stopping capability.
Turns out the sliding mechanism (pin) had no method of keeping water out… and it corroded SOLID. I had to replace the anchor (torque member). First time I drove it after putting everything back together, I nearly went through the windshield when I hit the brakes. My wife did the same thing.
Check for a sticking caliper. Not the piston, but the slide mechanism.