There's a spring loaded switch on that brake mechanism, that kills the ignition when the handle is released. The brake pad slows the engine down to a stop after it quits running.
This may be politically incorrect but I removed all that stuff from my Lawn Boy and replaced it with a single toggle switch located near the handle. Two wires from the switch to the two terminals on the ignition module and you're done. I also unbolted all that brake stuff from the engine.