Electrical - "off" circuit fried?

[Fonorono]

This is an 2007 argo frontier 650, Briggs/Stratton 13 HP.

So I misplaced my key so decided to hotwire it. On m first attempt, as the light was failing, I jumpered the red to the black wire. Mistake. Electrical smoke from the back of the engine, (firewall side.)
Then I properly jumpered from red to white & blue and then jumped the starter wire and got the engine to crank. But it wouldn't fire, even with quickstart.
I pulled a spark plug cap and there is healthy spark when I crank the engine.
Can anyone explain what the "off" circuit through the ignition switch does? Did I fry something in that circuit? Fuel solenoid or something that prevents it from starting?
It's really cold here. I have to keep charging the battery between my troubleshooting sessions.

[racerone3]

Even if you fried the carb solenoid (if equipped) it would fire up with starting fluid. check both cylinders for spark when cranking. briggs ignition is a magneto and should spark every time the magnet on the flywheel passes the ignition coil IF the coil isn't grounded through the kill circuit. if you don't have spark, try removing the kill wire from the harness to the engine. this is a small black wire a few inches above the starter. unhook it and check for spark again. If it only has spark on 1 cylinder (the one toward the front of the machine) you have fried a diode in the kill circuit. this is a wire located under the flywheel shroud. it connects the kill terminal of the 2 coils and the diode keeps them form "talking to each other" unless the switch is turned to the kill position

[Fonorono]

Update: I got the key, put all the wiring back together and it fired right up. Hurray!
When i turn it off, the key comes out but the engine keeps running. So it looks like i definitely fried the stop circuit. I killed the motor by pulling the choke and opening the throttle..
So while this works fine and leaves a nice shot if gas in the carb for the next startup, I'd like to repair/replace that stop circuit. I know the key grounds the coil when in the off position, killing the spark, but all the electrics are under the front of the motor, out of sight, and the wiring diagram just shows that black wire going to the ignition module. The spark still jumps, so the ignition module is ok.
The is no solenoid on the carb. .
Thanks racerone3 for your description. I will try to find that black wire, above the starter and see what's what.

[racerone3]

you will need to pull the flywheel cover to replace that kill wire.

if you have a volt meter, check for continuity between the kill stud on the motor and the negative battery terminal. you should have continuity with the key off and not have it with the key on. if you have a test light hook it to battery positive and probe the kill stud. should light up with the key off and not with the key on. this check will verify that the machines wiring is good

[Laird]

Nice easy to follow instructions. Learned about the diode the hard way. Didn't teach us that at Briggs "school" or maybe I forgot or missed it all together. Know about it now.

[racerone3]

Nice easy to follow instructions. Learned about the diode the hard way. Didn't teach us that at Briggs "school" or maybe I forgot or missed it all together. Know about it now.

only the vanguards have the diode wire. the older flat twin engine had 1 coil with 2 spark plug wires. each cylinder got spark at the start of the power stroke like normal, but also at the start of the intake stroke