Hi Vanessa, the first question is, what happens when it fails to start? Does the starter motor crank the engine, but the engine does not fire up? If so, by my reading of the wiring diagram, the seat switch is a possible culprit. The other safety switches (PTO and clutch brake pedal switches) seem to disable the starter solenoid as well as the ignition, but the seat switch only disables the ignition. However there are other things that can disable the ignition - such as a defective ignition module, a faulty ignition kill wire that is being intermittently grounded, a faulty PTO switch NC contact, or a faulty start switch.
If someone involved in this has experience working on fixing faults in potentially dangerous machinery, you can use a process of elimination. However this requires that some basic safety rules be followed explicitly. The first one is, only one person works on the machine and all others stand well back. The repairer temporarily disconnects the ignition kill wire (the yellow wire going to the start switch terminal no. 4). If the engine starts and runs properly with the kill wire disconnected in this way, but does not do so with it connected, the fault is in the kill switch circuit downstream of the start switch. Reconnect the kill wire as soon as you establish this - the machine is unsafe to operate without it, and you can't stop the engine except by grounding that lead anyway. If it isn't the kill switch circuit, tell us that, and tell us more about the symptoms, and we'll talk about other possibilities. If it is the kill switch circuit, we'll talk about how to check the various parts of that circuit.
Remember, that lawn tractor is a dangerous machine. One test at a time, only one person working on the tractor, that person fully in control of the operation and well-briefed, and think first before making a test. I've had to sit through briefings on how skilled tradesmen in a very safety-conscious company managed to kill each other when two of them worked on a job successively, and something went wrong with the hand-over procedure. Do not let this happen to you.