Symptoms are just like mine earlier this year so I will relate my process when my 3600D would not start this spring.
****MAKE SURE TRACTOR IS IN NEUTRAL****
Since you have 12+ volts lets assume battery is OK. If you don't completely trust the battery put jumper cables or a booster box on it.
Jumper solenoid at starter if this forces tractor to turn over it is somewhere in the small wires.
It is simple to isolate the neutral switch. Follow wires that come from the transmission cover and remove lower cowl cover. Pull 2 plugs that are behind cover and jumper the ends together. If tractor starts now it is a bad neutral switch.
Since you now have the cowl off it is simple to remove the key switch and use a volt meter to see if voltage is ending up where it should to isolate which small wire it might be (or possibly a bad switch).
Now my experience this spring was that the starter would not spin when jumpered so i pulled it and took it over to my local rebuild shop and they tested it as good. Had 12 volts at the right places so started looking at the rest of the start circuit and had voltage everywhere it should be. Bought new battery cables and replaced and everything worked fine again. Spent a couple of days pulling my hair out and bugging the folks on these boards before I broke down and picked up new cables. Cables were corroded enough that they would let small enough current flow to read voltage, but when starter actually wanted to engage they would not let enough current flow (My simplified layman's understanding).
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