This is an interesting early electronic ignition system if this is the original system.
There are 2 electronic ignition modules, one for each side. The kill function that comes from the kill switch, tether switch, and ignition switch has 2 separate 'kill wires' that run from each of those switches to the engine harness connector and then to the 2 ignition modules. It sounds like one of those 2 wires is grounded somewhere on the chassis harness side of the engine connector, or is shorted to the ground return wire.
On the engine side, both of those kill wires to the ignition modules are black, and the ground return is brown. On the chassis side, one kill wire is violet, the other is violet/white, and the ground return is violet/brown. Trace out the violet and violet/white wires to see if one is grounded. If nothing is obvious, then disconnect each of those 3 kill switches and see if one of them has a permanent short from violet or violet/white to violet/brown.
Wiring diagram is here, on page 193 of this .pdf document.
https://vintagesnow.com/Ski-Doo_files/1980_Shop_Manual.pdf