Hi bertiesme. Thought I would bring your pic back up to the top the current thread. Of course, you have a single play, belt drive. My mistake.
If the long speed selector rod is nice and floppy now on each end that is good as the only thing that moves this rod and selects the speed is the small red spring pushing with friction against the vertical rod the long speed selector rod is connected to. Your red line in the pic cuts right through this spring at about 1 o'clock. The spring is small so that is why the parts it pulls against must be loose and floppy. I think this portion on your machine is working correctly.
Seems as if the belt drive has just enough strength to turn the large cam to return the tonearm at the end of an lp but not enough strength to make it lift and move towards the lp to start with. Makes sense because it must move more parts at the beginning than at the end so something is sticky. Check all parts within your red circle that have a spring attached or if there is a spring missing. Also, I assume that you removed the large cam entirely and cleaned and re-oiled it. All old grease and oil must be removed from the plastic parts attached to the large cam. These trip pawls must be dry to work right.
The part just below the right split ring can be pushed by hand to the left as mentioned before. That also moves several parts one of which is the slider that slides over the white plastic part, quadrant you mention. These parts must move and return via a spring to their normal position very easily. Fix those if not.
Your machine has a different trip pawl rod system, mentioned above, than the 100, so I cannot help with that but I assume that it trips and engages or disengages a plastic part on the large cam. Check that and the part it is attached to.
If you could set your machine on a piece of glass or prop it up in some manner so you can see the bottom cam and parts operate and if the cam actually turns when the auto function is engaged. Sounds as though the cam cannot be turning if the tonearm will not move towards the lp. With the power off, pushing the auto function will still engage that function so push the auto toggle and then turn the platter by hand. Now, does the tonearm move correctly. If not, some part is not engaging below. Is there a blowup parts list for the 100SP here in the library? I will look myself to see the parts that are different from the 100.
Is the belt you are using new, old or possibly stretched so it is slipping at the time when the auto is engaged and can't turn the cam. Could be a simple problem here with one missing or stretched spring or a sticky pawl. Each spring on the machine has a function to engage, disengage, etc.
Yes, you are on the right track if any part does not slid easily.
On the left hand of your circle, there is a tube the tonearm wires emanate from. Check the arm with the 2 holes the tube passes through. Has a set screw that must be tight as this turns the tonearm.
Hope this helps, best regards, chan
PS: Yes, I see the 100c. Not interested but thanks for the heads up. I may settle for a SB such as yours but I really want just a plain S.