What you describe is also a symptom of a safety interlock circuit problem, which is not uncommon. All the other possibilities mentioned above could also be the issue.
You could eliminate things other than the safety circuit if you bypassed the starting solenoid but this can be very dangerous, especially with a gear tractor, that could be left in gear and will start moving immediately. I suppose doing this would not be recommended.
If you have a voltmeter you could measure the voltage on the terminal on the starter where the voltage that activates the solenoid is connected. You will likely find it drops pretty low when the key is turned to start, and the solenoid clicks. But this only indicates a possibility, as the voltage everywhere could be doing the same thing at that point.
If you see the voltage dropping at the solenoid safety circuit terminal, the next check would be to check the battery voltage where the large lead attaches to the starter solenoid, and see if it is also dropping when the key is turned to the start position. If it is also dropping the problem is one of the things mentioned above and likely not the safety circuit. IF the battery stays up when the starter clicks, the problem is likely the safety circuit.
If it turns out to be the safety circuit, which is not uncommon apparently, there is a solution involving using a relay between the safety circuit and the starter solenoid. But that is something to worry about after you isolate the reason for the problem you are having.