A Spartan is a Singer, the cabinet looks to be a Kenmore or White.
I don't know what exactly the "weird oblong dohickey" is, but the cabinet wiring as you drew it looks like a Kenmore or White machine would work in it.
The motor is the Singer coupler, two wires go to the foot controller, two to the wall socket. They are or were wired in the square bottom of the motor so it would work. The Spartan did not have a light.
The connector block with the BLK, RED, WHT slots would normally plug onto the motor of a Kenmore or White machine.
Your conundrum can be fixed, but from here I can't tell you exactly how to do it. You've got two different systems there.
Joe