According to the diagram, there are four safety switches. Seat, reverse, clutch, and PTO. According to the wiring diagram, the yellow wires are for the starting circuit and the red wires are part of the reverse over ride circuit. We will concentrate on the yellow wires. To start the engine current leaves the key switch and goes to the clutch safety switch. Brake pedal pushed down safety switch closes sending current to the seat safety switch. Sitting on the seat closes the safety switch sending current to both the PTO switch and the reverse switch. If the reverse safety switch is working properly, with the unit in reverse the circuit will be grounded and the unit will not start. If the unit is in neutral the switch is open and current will go to the PTO switch. If the PTO switch is working properly with the PTO off the current will go to the solenoid, allowing the engine to start.
After I went through the diagram you listed ( I know this is not your unit ) i think you must also have weight on the seat to start the engine. I hope this makes sense to you. It is much more difficult to troubleshoot this when I cannot see everything you see. I will continue to work with you to solve this problem, as long as you want me to. Sorry, I know that I repeat my questions, but I am working through your eyes, and with other posters, and I forget which units I am talking about. Please bear with my short comings, as some on this forum don't feel that I know what I am talking about. Would you also check that unit brand name and model number. I cannot find it in any of my MTD material.