After the problems I had with the last batch of boards, I went back to the drawing board. Well, back to hackaday.io anyway, where I found #Business Card Gamepad, which I figured might be a solution to my USB Type-C receptacle woes. I found the relevant drawing in the USB Type-C spec and drew it as a KiCad footprint, which I then used to make a new version of the board. Since it has to be 0.6 mm thick now, I couldn't order this board from OSH Park, so I went to Dirty PCBs instead. The quality is quite good; the boards aren't dirty at all.
I originally considered gluing on thin pieces of single-sided copper-clad FR4 to act as shield pads. I tried this on one board and found that it made no positive difference to functionality or robustness. Securing them well enough was difficult, and excess glue caused some problems. So, I ultimately decided to leave them out.
The new board works just as well as the older board revisions, and is much easier to build because I only have one connector to solder now. I just listed them on Tindie, at a reduced price due to the lower material cost. Hopefully this will be the end of the PD Buddy Wye supply problems!