In my experience, if they don't include the conductor in the connector, there will generally not be a wire for it in the cable. So replacing the USB-C connector is probably not going to add any functionality.
I have a stack of USB-A to USB mini/micro cables that are a bit like this. Two conductors, only do charging. So if I replace the connector, there's no wire to connect the D+ / D- lines to. It's quite possible the manufacturer of your cable had some similar or otherwise reduced feature set in mind, and wanted to save 5 cents of copper.
I also have some cursed cables that have normal ports, but only two wires inside the cable (power/ground), or are otherwise out-of-spec in ways that make the data lines mostly fail. These have wasted quite a bit of my time over the years.