[ I came across something related to this on my KoS very recently. Replying here in case it may help someone in the future ]

Yes, the brown cable that is tied to the Green one on one side should be hooked up the the switch where the cable is cut on the other side (when reaching EOS, it should connect the green cable to yellow/black cable).

Seems like the previous owner did that to bypass the broken switch (as you mentioned).
While it "kind of works" (top left & left flipper will fire) you will have 2 weak flipper (as the power is send to both at the same time, therefore solenoids will be much less stronger).
The point of that 2nd switch is to have the first (bottom left flipper) go full power, then when it reach end of stroke (the bottom bottom left flipper switches to low current, connect high current to the top left flipper) have the top left go with full power as well.

While I did not notice while playing, the bottom left fire first THEN the top left fire when the first one is done (EOS).
The two flippers movement are happening very quickly one after the other but they appear to be occurring at the same time.