You are correct. The operator presence safety circuit is probably killing the engine.
The wiring schematic for the mower is on page 31 of the owners manual. If you need an owners manual, you can download, view, or print a copy free at [managemylife.com
When you are not sitting in the seat and the clutch/brake pedal is released, the seat switch and connector supply ground to the clutch/brake switch and it makes a circuit that grounds out the ignition module.
The seat switch or the seat switch connector is usually the problem when the engine dies when you release the clutch.
The seat switch connector has a shunt built into it that closes the circuit if the connector is unplugged from the seat switch. Be sure the connector is securely connected to the seat switch.
You can test the seat switch with an ohm meter. The seat switch contacts should be closed when the switch is not depressed, open when the switch is depressed.