I will also look into this in the weekend and try to measure around the “auto start” circuit and compare the difference between the good speaker and the bad one. Please find the service manual attached which include circuit schemes and block diagram as well as some instructions that can help. It mention that you can short-circuit the C83 in order to block the autostart so this capacitor could be the error cause. I will check the electrolyte capacitors because these can show strange behaviours if the are old.
If you want to make your own RCA to DIN cable, connect pin 4 to maximum 5V and pin2 to ground. 
Please note the the RCA-DIN connector will be mirrored compare to the picture above.
Please keep me informed if you make any progress.
/Carl