Welcome to our little group. It is a divide and conquer everything knda deal. If it were me I would connect up the negative lead as normal, the take the other + lead, on very low volume, and start jumping components that are hooked up to the + wire traces/ wires. I have a feeling that it could be a failing reset switch or some of the caps going bad after a 20 sec warm up. Being an older model Pioneer it may not have a re-set switch but a fast blow glass fuse. Glass fuses either works or not. Reflowing the solder joints can help as well.
Doing this can help you know which driver or crossover board component could be failing. You can check your drivers with a regular ohm meter as well. It is my rule of thumb with Electrolytic caps, like the ones in those, and over 20 years old, to replace capacitors with near the same uF rating or near as, rating over 100 volts. You can twist the cap wire leads together to obtain a closer uF rating too.
All, If you need to replace the caps, you can use new and longer lasting Poly or "Film" caps like Parts Express "Dayton" poly types. Another option is to replace just the caps in the low end with Elecctrolytics and Polys in the upper end tweeter circuits.
Best of luck anyway. Intermittent shorts can be a real time eater....If I can help, feel free to PM me and I'll do what I can..
DC