Thank you. I didn't know there were additional pins in the USB-C connector. This site cleared up a lot for me.

I guess a USB-C to USB-A cable would only have the 4 conductors, but a proper USB-C to USB-C cable would include additional conductors.

Like you said, I need a proper USB-C breakout board, like this one that exposes the additional pins (this one even includes the proper resistors for cc1 and cc2).