cleaning connections is worthwhile, critical even, all it takes is a bit of road grunge and there can be a bad connection - could have been the extra non-original wire doing something too
yes that harness should have all the correct stuff you need, and all the suppliers are about the same price ... remember that the assembly manual was done in 55, when the V8 was primarily available in big trucks - the 6 had the foot start, so a 2 position switch, the V8 had solenoid start, so a 3 position switch - later [and aftermarket] switches had the ACC position as well, so 4 .... some wiring has individual connectors for the switch, some had a single plug with all the wires, any for a V8 [with 'start' position] can be made to work fine
Bill