So I got my screwdriver out to see what the issue was. This is what I found. The connection wasn't being made. These two pieces of metal were supposed to make contact when the stylus was positioned. Here, in this picture, the stylus is at rest and the connection is not being made.
What really needed to be done was to eliminate that part as a whole, because it really was not a great design and after so many years of use, it had pretty much stopped being reliable.
So I cut off the leads which led to that mechanism. It was one single red wire leading to a DC motor. The return wire went directly to the motor, and the positive wire went to the problem.
I re-routed the wiring so that it went directly to an SPST switch. That switch was now the new "on/off connection point for the turntable.
I can now turn on and off the turntable with that switch, which is much more reliable than the old way! And I can also put my turntable back together again and listen to music.