Reads like someone(previous owner) installed a kill switch, if that connector is OK, because there is nothing in the seat electrics that would cause the engine to die without blowing a, or some fuses, and it certainly wouldn't restart with blown fuses that killed the engine.
Most likely cuts fuel pump power since it is easy to get to from inertia switch in passenger footwell.
Look under the dash for a toggle or push button switch, sit in drivers seat(engine off, lol) and feel around, it should be accessible by driver to reactivate engine.
The fact it restarts when drivers seat is empty means factory electrics shouldn't be involved as shorts blow fuses.
As a long shot I would check to make sure there is a ground strap from engine head to firewall and both ends are tight, this is the cab, computer and instrument ground, if computer loses ground(power) engine will shut off.