You're correct, the ball count unit is a step-up/step-down stepper and not a step up/reset stepper.
Sometimes a rough looking rivet will conduct ok if you adjust the wiper arm slightly into a different position. If you are to repair, use a dab of silver solder (heating the rivet from below). Do not use regular lead/lead-free solder. You will need the tensile strength and better conducting properties of silver solder. Note that you will likely have to desolder the wire connected to the rivet to do this. I suspect that it is conducting ok, since your lifter functions when you depress the pin to close the switch.
The pin under the lifter can be hard to adjust sometimes. Occasionally, the problem is actually the lifter cam/arm. It will not travel all the way down, allowing the ball to rest on the pin. There should be a set screw adjustment about midway up the arm of the lifter. Loosen this, then tighten in a different position. If the other problem happens (overtravel), do the same thing. Expect another 20 minutes of farting.
You've already verified the rest of the lifter circuit, so your lifter problem should be resolved when that arm drops the ball into the right spot.