Has to be something with the switch or the wire. Basically all the switch does is make the wire from the engine touch the frame and ground out. If something chewed the wire it will never connect the circuit. You might have to take the back cover off the engine and maybe even the flywheel to get a look at all the wires.
Do you still have the interlock switch hooked up (right handlebar lever)? If it's hooked up and neither side handlebar lever is squeezed, trying to engage the auger lever on the dash should kill the engine. I have a 924038 but removed all that wiring, switch and rods inside the tractor. By switch, do you mean the keyed switch up in the dash, or the thing inside the tractor body? The key switch is pretty basic, just 2 wires.
How do you know "the switch is fine?" Simple test would be leave the key in the "off" position and see if the engine will start.. Did you do any repainting or degreasing around the lower recoil shroud Under the carb area? Did you try engaging the auger without squeezing the handles while the engine was running? Here's my old switch from a 924038. If everything else checks out, inspect the crimp material on both wires at both ends for corrosion or breaks.