I've tried to bypass all the safeties before and didn't work for me, ended up pulling the relays all together and rewiring my machine without all the safeties and ended my headache - pretty sure the relays are in Series so each one needs to satisfy the next etc.
Jump your solenoid and verify the starter will engage first then work back from there - ignition switch etc.
If jumping the solenoid does not work , it's probably a ground.
Odd that it stopped while mowing though , will the machine turn over by hand ?