When working as an electrician I learned years ago to always start my troubleshooting from the beginning and proceed logically through the circuit step-by-step. Verify the integrity of the circuit, beginning with the source (battery), connections (cabes and wiring harness connectors), and finally any controls (safety switches, solenoids, etc.). To reduce future problems with the starting system, now is a god time to baseline the system. I assume it's already been suggested to test your battery. Verify proper electrolyte level. Consider testing electrolyte specific gravity with a battery hydrometer. Consider taking it to your local auto supply house and requesting they load test the battery; you may read 12.6v w/ no load but it may not have sufficient cold cranking amp capacity under load to correctly operate the starter or turn the engine over due to one or more bad cells. Regardless of all else, be sure to disconnect and clean to shiny metal all battery cable terminal connections (battery posts, solenoid posts, starter posts, anywhere the full-size battery cables terminate). Use a fine wire brush, battery post cleaning tool or fine emery cloth. Consider appying a dab of oxide inhibitor to all these connections. Your auto parts or local home store sells this in the electrical dept. Consider disconnecting and reconnecting all safety switch terminal connectors several times to 'abrade' them to ensure a good electrical connection. A small dab of inhibitor here is a good idea too, just remember you're seeking a thin layer to prevent corrosion and oxidation.
I haven't read all the posts so I'll ask this directly: When you use the key I presume nothing happens? If correct, then after completing the steps above, use a multimeter to measure voltage at the battery posts. And then read voltage at the starter terminals when turning the key to start. Do you measure the same battery voltage there as well? If so, then it seems your safeties are functioning correctly, and I'd focus on the starter. If not, then it appears you may have issues with the wiring, terminations/terminals, and/or one or more of the start safety switches (seat, clutch, HST treadle, etc). Find the wiring diagram for your model tractor as a guide to troubleshooting the circuit.
Once the battery checks out or is replaced and the wiring and switches are verified, it seems to me that if the starter spins when you use a remote switch connected directly (bypassing all safeties) but the engine does not turn over then one problem lies with your starter pinion gear not engaging the engine flywheel ring gear. Consider rebuilding or replacing your starter.
Report back once you've tried these basic troubleshooting steps.