There are two lots of "safety " circuits on your mower.
One stops the starter from cranking
The other stops the plug from firing.
Shorting out the solenoid defeats the first but not the second, hence the unburned fuel smell.
There should be 2 small terminals on the starter solenoid.
One is + the other -
The + comes from the ignition switch "S" terminal then goes to the PTO switch and some times the parking brake or lap controls then down to the starter solenoid
The - goes through a convoluted path involving the seat switch, PTO switch & lap controls in varying combinations.
So take a voltage reading from each of the small terminals to earth with the key in the start position.
The one that goes 12V is the +
If you get 12V between the terminal & earth then try between the two terminals.
No voltage then check for continuity betwen the - and ground.
From here on it is a matter of working backwards through all of the switchs & relays to find the one that is gving you grief.
I have a box full of jumper leads with 2 Male or 2 female terminals on them which I use to bypass each switch, one at a time till the culprit is found.
Remember some switches will make/break both a + connection and a earth connection and some of the earth connections are also for turning off the spark.