Depends on your definition of a full system. A basic non rehab setup has only these:
1. Battery to controller. (batt terminals, often 6mm ring terminals, and 2 blade connectors on the power module.) Now it has power!
2. Motors. Mostly come with connectors. These plug into the power module. If they dont, use automotive blade connectors... Now they do.
Thats all. Joysticks use their own bus type cables to connect to the same family of controller. So thats not involved here.
If your chair has more wiring than this throw it away.
On controller...
Motor 1 is motor and brake.
Centre, battery...
Motor 2 is motor and brake.
All just different sized blade connectors. In a fancy plastic plug casing that you dont actually need...
Crimp, solder and add heat shrink. Or use the ones with a clear hard polythene cover.
Large ones are 10mm small are 6.3mm I think from memory.