It is not a good thing to leave the key switch in the on position with the engine not running particularly if the clutch switch is still in the on position.
What kills the battery is the draw of the clutch and the draw from the engine charging system which is attempting to turn the engine over thru the stator. When the keyswitch is in the off position, these systems are then electrically isolated from the battery.
All this being said, you can get the model and serial numbers from you machine, go to the Scag website and download the manuals for your specific machine. You can trace thru the wiring diagrams and by the process of illumination check each component starting at the keyswitch.
I would start at the engine first and as you had mentioned, jumper across the start solenoid or direct from the battery positive to the starter to make sure that it will turn the engine over. Sometimes this takes a separate battery cable to do. Bear in mind that possibly 100 amps or more will be drawn from the battery and some sparking will happen. The starter solenoid may be engine or frame mounted and the operational side, the small wire, may be jumped and this will operate the engine cranking system more safely for testing purposes. If the solenoid is frame mounted and has two small wires connected to the terminals on the base, then one is a ground. Do not connect a positive jumper to this, the other terminal is the positive, jump the positive terminal.
The electrical system on Scag SSZs is fairly straightforward and troubleshooting should be easy provided that you are skilled enough and have a multimeter for electrical testing.
Initially, make sure that the travel levers are fully extended outward, the parking brake is on, the keyswitch is in the off position, the fuses are still good, the PTO/clutch switch is in the off position, the battery is fully charged, the battery cable connections are clean and tight. There is a seat switch that can be operated by hand with the seat in the up position. So check it out and let us know what you find. If you do not feel qualified to do this type of troubleshooting, then find a quality servicing Scag dealer. And as always safety first!!!
Mad Mackie in CT:biggrin::laughing: