You can hear the fuel pump kick on, after I turn the key to On.
That isn't supposed to happen. The pump on a 2V Brick only starts when the fuel injection relay is transmitted signal from the Hall sensor when the engine is cranking,
not before. The wiring from the ignition control unit to the fuel injection relay, the fuel injection relay, some weird wiring from the previous owner, or the ignition control unit itself all could be possible sources of this condition.
Regarding Harris's video:It's possible the piston rings could be seized and lubricating would release them, but that doesn't resolve the faulty behavior of the relay. Are the spark plugs wet or dry after an extended round of cranking?