These controllers will come up used sometimes. Softenerparts has parts if you can trace the problem to a single part.
http://www.softenerparts.com/Autotrol_Repair_Parts_s/2.htm That site is also a good place to download the manual.
It may just be time to replace the controller, or maybe the whole thing-- particularly since the resin does not last forever and the current size may not be right. I am still using mine however. I don't know the age and I don't know how old the resin is. Still working. A new softener would regenerate based on demand, so it would use less water probably compared to a purely softener.
If you just replace the gear, I wonder if it would be a good idea to lubricate something to reduce the load on the new gear. I don't propose to say what to lubricate. I guess the plastic-to-plastic bearing surfaces and maybe some things on the cam.
See
http://www.softenerparts.com/440_440i_Timer_Repair_s/63.htm The way this thing appears to work is that the gear teeth on the red knob piece, but inside the cover have gears that drive the red counterclockwise.
There is a gap that corresponds to where the red thing is pointing down to the service position.
The skipper wheel pin (when in) pushes a molded protrusion (
A on photo -- would that be called a dog?) in the red piece to rotate it CCW past the gap.
I am not totally sure why you have to push the red in to move the red thing, but the skipper wheel can do it. I think maybe the red gear moves just enough to get the gear touching, and the skipper wheel continues to apply pressure as the motor moves the gear. If that is correct, you could apply torque by hand for a period of time, and the arrow would slowly move to Start.
So I was wondering if something could be done to reduce friction. If you remove the screws to take off the face plate, take photos at each stage to help you get it back together. The exploded diagram helps a lot, but I would still take photos. To just replace the white gear does not involve taking off the face plate. I wonder if a spray lube could be useful.
That white thing you think is a lever is just a pin that the timer in place IIRC. Push the flag to left or right to lock pin. . Point the flag up and pull the pin out to unlock the timer. Then you can lift the timer up and out to replace it or to work on it away from the softener. The power cord has a strain relief. If you rotate that, you can line the strain relief to where it can slide out of the slot to totally detach.
To set the softener to regenerate tonight, if it was working, you would push the Day 2 pin in. I think you already have your pins in, so it seems probable that you have the mechanical problem that Ditttohead suggested, or some other mechanical problem. I don't remember if you said the clock continued to keep time.