By you're testing of your switch it sounds perfect. They are a much more complicated switch than you think. Small engines do not require 12v+ to run, (unless modern not youre case.) the switch will only send power to S for start, L in on for lights/accessories. The switch has G which is ground, so in the Off position it will link M (magneto) to ground to kill the engine. otherwise in on/start that should be open.
Since you appear to have a multimeter, start tracing voltage, it will start at the battery and go through a fuse to B. if you have power there, start tracing S. It will go through 2 of the switches before it reaches youre solenoid. The idea is you can't crank unless you're blades are disengaged and foot is on the safety button so you cant drive.
The seat switch will kill the engine, the foot switch will stop it cranking, and the blade switch will stop it from cranking as well as kill the engine if you are not on the seat.
