The ignition on the 318 is enabled for starting when the ground speed lever is in the Neutral/STOP slot
AND the parking brake is set
AND the PTO switches are off...these are the same conditions that allow the starter to crank the engine, and the simultaneous satisfaction of these three conditions also provides a bypass voltage to the TDCM on terminal X23, pin 7 so it can ignore the seat switch. Here are the relevant wiring excerpts:
For the engine to remain running when the PTO is switched on (and/or the ground speed lever taken out of the stop slot, and/or the parking brake released...) the seat switch MUST be closed by the operator being present and the wiring and fuse F2 intact and providing voltage through it to the TDCM terminal X22, pin 9.
Sorry for using multiple wiring excerpts -- but posting the entire schematic here results in a drawing that is too small to see well for discussion purposes... It is unfortunate that the drawings do not each show all the connections to key switch terminal S2...but the lower of the three excerpts comes closest to showing how these fragments interconnect.
I am pretty sure that your seat switch and it being "disconnected" is the actual problem -- my recommendation is to restore your tractor wiring to its factory condition so it is easily maintained and understood by yourself from the Deere documentation -- and that would also help other members of this forum assist you directly rather than 'guess' what hacks other prior owners may have done to your tractor...
Chuck