I've seen something similar to this before although not in the way I was seeing with a recent 4 switch console I was servicing. As part of diagnostics I will use Paul Slocum's excellent Test Cart program as it should some primary colors, shows the current state of all switches minus power of course, but also has a basic graphical view of each controller and small block on the bottom that will move left/right when you plug in paddles to test those too. So all in all a nice utility to know that all controller functions are working properly on the console.
Well, on this one, player 1, player 2, and player 4 paddle controls would move from left to right and back turning the paddle as you would expect. But player 3 paddle control would just site on the left side, and then after a point when turning the paddle, it would suddenly just be on the right. No movement of any kind. Just one sec on the left, and then next thing you know it is on the right. So it was acting more like a digital control vs analog. It wasn't the paddles since I used the same set to test player 1 and 2 and that was working fine.
Going through the service manual will yield some interesting stuff to help isolate this, but unless you have the diagnostic controller plugs and the 2.6 diagnostic rom, you aren't going to be able to see exactly what you need to see. But lets review that..
If you have the diagnostic plugs, plugged into the controller ports and the diagnostic rom up and running with the controller matrix screen up. Then you use an Oscilliscope to probe the paddle lines off the TIA pins 37,38,39, and 40. They represent player 4, 3, 2, and then player 1 on pin 40. Well, what you should see on your scope if you have it set to the right settings, is something like the picture below from the service manual:
However, when I probed pin 38 that is for the player 3 paddle line. I was getting a flat line. Well, actually I was showing a flat line of about 1v but the point is...not pulse like you see in those pictures. (And btw...I was seeing that same pulse line on my o'scope for the other paddles).
Well honestly there isn't much in the way of electronics from the controller port to the TIA where the paddles are read and handled. In fact, there is really only 1...just 1 component in the middle of the mix from the controller port to TIA. At least on the 4 switch and above units this is the case. That one component is usually a small ceramic disc, or poly capacitor that doesn't usually go bad. So I first checked that the traces from pin 5 of the player 2 controller port to that cap (C220) was good. It was, and then checked from the cap to pin 38 of the TIA. That too pinged out good. So I went ahead and replaced the capacitor just to see if anything changed. Sadly.... no.
What did fix it?
Well, if you've gotten this far and read my description of the very simple circuit from port to TIA... it should come as no surprise that is was the TIA itself. This is even more sad considering how rare these IC chips are now becoming and there isn't any projects I'm aware of to make new ones or something to replace the TIA.
But yeah... if you find the paddle lines aren't working, chances are that it is the TIA chip itself that has failed if the actual traces are good. Apparently this was less of an issue with earlier 2600s as they used buffer ICs to help control this and therefore the TIA was more protected. Just more cost cutting at work as the console lived on...