There is a module in the cab that controls all the safety. Mine is a hydro as well. There have been more than one of us that this module has gone out on. Those of us that have had a bad safety module pulled our hair out checking switches, fuses, relays etc. The thing about the safety module is that if it is bad it will not prevent the tractor from running once started, it just prevents it from starting with the key.
Here is the kicker on the start relay under the hood -
1. The safety module tells the ECM whether it is ok to start or not. Tractor must be in neutral, seat switch says you are in seat, brake depressed, brake lever flipped over so both brake pedals locked together, PTO disengaged, neither forward of reverse hydro pedal depressed. If all this is met the safety module sends the signal to the ECM that it is ok to start the tractor.
2. The circuit that is completed is the Negative on the relay. So the way it works if the module says it's not ok to start then the ECM will not complete the ground circuit on the start relay when the key is turned over to start.
3. If you jump the negative on the start relay under the hood to a frame ground you can start the engine with the key - caveat is that it will have bypassed every safety in place to keep you from starting - so it could start with anything listed in #1 above in the wrong position.
When mine was going wonky I could unplug the safety module and plug it back in and the tractor would start. Dealer replaced it with supposedly updated one under warranty.
If you read through that thread I linked to the second page of it you will see another poster attached the wiring diagram you will need to figure this out.