USB 2.0 cable to USB C output
As Tom said, both black wires are grounds. You should be more concerned about correctness of auxiliary pins on the Type-C receptacle, CC1 and CC2 in particular, otherwise it won't work.

Fortunately, the breakout board does already have proper pull-downs, and a proper cross between D+ and D- wires. 



So, you are lucky, and your rework should work. I am not sure although if you will be satisfied with thick, stiff, and expensive Type-C cable, which won't hold in the receptacle and fall off under nearly every movement of your device.