You can fiddle with a switch forever... If you suspect it, install a new one, but it could be several other things.
Brake switch, fuse, wire chewed by a rat, broken wire, starter solenoid, etc...
On the switch;
If you find the 12V supply, then turn it on and you should find a constant 12V on another terminal.
Then in the "start" position" most will make the "S" terminal hot, then this goes away when the switch springs back to run.
You can short across the 12V and the "S" terminal if you can determine where this is.
You really need a wiring diagram which will be in a technical manual, and then you may still be confused..:confused3: