The way I understand the solenoid is that both coils of the solenoid energize to overcome the spring. After 3 seconds (timer module controlled), the pull-in portion relaxes but the hold-in portion is energized the hole time. The hold-in coil is not enough to overcome the spring by itself, thus the pull-in coil must be working... I would think that this is working on my tractor or the fuel wouldn't turn on at all. But I don't know why it starts only on ether even when warmed up. I don't know how many hours are on the tractor because the hourmeter is blank but if I had bad rings or something I would be getting blow-by. If it was an injector issue, I wouldn't think it would run well or it would smoke. Would there be something that boosts fuel or changes timing, etc on engine start that is malfunctioning? I wonder it the shutoff is sticking and the jolt of the engine starting with ether is enough to open it all the way?
Pretty sure the problem is not bad rings, an injector issue, fuel boost, timing, or even a sticking fuel shutoff solenoid valve. Unlike the newer fancy tractors that are computer driven (down to the fuel injectors), your tractor is pretty dirt simple and just needs fuel to run.
If you don't want to buy a timer module, or remove the K3/K4 relays and use the jumper method for testing I described in my previous post, you have 2 additional options.
You can unplug the fuel shutoff solenoid, apply 12V to terminal A, ground terminal C to activate the hold-in coil, then briefly ground terminal B to activate the pull-in coil. Start the tractor with the key as normal, then remove power from terminal A (or the ground from terminal C) to turn off the tractor.
The last option is to remove the 2 bolts and the fuel shutoff solenoid valve from the engine. It should then start, but you will have to either re-install the fuel shutoff solenoid valve, or block off the intake airflow to shut down the engine.