I have attached a diagram for the 5X10 starting system, which is close if not the same for the 5X20. You will see that the path to the fuel solenoid is different when the key is in the run position and when it is in the start position. Since it cranks the safety switches are ok. I would say the bypass start relay is bad or contacts need cleaning. If the bypass start relay is not working current flows of it onto the fuel shut-off relay, which cuts power to the fuel solenoid.
If needed I can provide you with a write-up on how the starting circuit works.
On my 5210 the relays are located under the left cowl. Before pulling out the relay, I suggest you disconnect the negative battery cable either at the battery or where the cable connects to the frame.