The only way I can see that this could work is if the original turn signal switch INTERNALLY ...

a. provided a ground where the 20-DBL (20 gauge Dark Blue) wire connects while the turn lever was signaling a left turn and flashing the 20-LBL (20 gauge Light Blue) wire.

b. provided a ground where the 20-LBL wire connects while the turn lever was signaling a right turn and flashing the 20-DBL wire.

c. provided the same voltage (either 12v or [more likely] ground) to both the 20-DBL and 20-LBL when the switch is in the center (no turn) position.

What would happen is that voltage would switch directions through the single turn signal lamp on the dash depending on which direction you were signaling. As long as one side of the lamp wasn't grounded this will work. All the other dash lamps show a ground, but the turn signal lamp DOES NOT.

Pictures below are of your wiring diagram with a little color added to make it easier to trace the wires in question.

By the way, this should answer your original question "what is the purpose of the brake light ground wire being in the turn signal switch?"
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Last edited by Bill Hanlon; 08/03/2024 4:08 AM. Reason: grammar

'57 GMC 102, Original 347 V8, HydraMatic, 3.08 rear gear, added A/C, disk front brakes, HEI, AFB carb, '98 Honda Black Currant paint. T-boned and totaled 10/12
'52 GMC 152 Stake Bed, Original 228, SM420, added A/C, HEI, disk front brakes, '67 Chev 3.55 rear gear. Gets used as a real truck.