There is a yellow wire with red and green (or possibly violet) tracer colors that runs from the MPEM to the starter solenoid. Unplug this wire from the starter solenoid and connect the wire to a voltage sensor of some type. When you press the start button, if the MPEM decides all is well, it will put a +12V signal on this wire to energize the solenoid. If the wire shows 12 Volts when you press the starter button, then your problem is in the solenoid or the starter or both.
I suspect that the wire will not show 12 Volts due to the other diagnostics and replacements you have already done. If this is the case, next you must determine why the MPEM is not sending a start voltage to the solenoid even though you are requesting it via the start button. This breaks down into two parts. Either the MPEM is "bad" (unlikely) or it is not receiving one or more of the inputs it needs in order to energize the solenoid. In no case should you willy-nilly replace the MPEM without absolutely proving that it is bad.
The two most likely candidates for why the MPEM is not sending a voltage signal to the solenoid are the neutral switch circuit and the DESS, in that order. At the MPEM, the wire color from the neutral switch is red with a white tracer. The way the neutral circuit works is to send a ground signal to the MPEM. So, you must disconnect the neutral switch wire at the MPEM, connect a continuity meter to the wire and confirm it goes to ground with very little resistance. If it does not show a ground, then the task is to find out why and correct the cause.
The safety lanyard circuit is a real trick diagnose and I'm not real sure I know how to do it properly without a factory diagnostic tool. However, if your unit works like other similar units out there, you should hear two short beeps when you plug the lanyard in. If not, there is a problem with the DESS or possibly the neutral circuit. You need to get a free copy of the operator's manual as it gives you a list of the warning horn signals and what they mean.
One last thing to check and, really, it should be first on the list. Check all the fuses in the MPEM. There is one 5 Amp fuse in particular that powers the MPEM.
Let us know how you're coming along.