What you put power to and is clicking is the fuel solenoid. The device (two black wires going into connector) is the thermistor sensor/temperature switch (thermistors are temperature variable resistors used as sensors). For now you can disconnect it, it is part of the "cold start advance".
I cannot hear my fuel solenoid click the only way I know it is working is to put a DC voltmeter on it. The back terminal is the positive.
At this point it appears the fuel solenoid is not getting power. A very common problem with the 3029 engine is this sensor part # RE503242 shorts out and blows the fuse for the fuel solenoid. I know you said all the fuses look ok, but I ask that you look again. On your tractor I believe that would be the 30 amp key switch fuse.
I assume member joecdeere sent you the tech manual can you post the page that shows the starting system operation, let me know if you do not know how to do this. If fuses look good the problem maybe with a relay, but I need to see the starting system picture to make sure.
If you do find a fuse bad, replace it and leave the sensor disconnected and try and start it. If you hear the fuel solenoid click, but does not start, you will need to bleed the system.