In the implementation, the design uses a 'card edge' type connector with the whole PCB fitting into the constraint of the thickness of the center blade of the USB-C connector. So when charging, the board slots into the cable-end connector, and when not charging there is only the PCB. The PCB has cut-outs for the cable-end connector to fit into the PCB -- so like you're saying, but even thinner.
Thought the same but from what I can see those are somewhat uncommon, presumably you are unable to affix them to the board as securely as the typical connectors
Like this one:
https://www.adafruit.com/product/5180?srsltid=AfmBOopEIapZEq...
But with supports along the sides that could solder one the top and/or bottom of the PCB. You'd have a notch cut out for the connector.