Ran into the exact same thing last month with one of those cheap F411 boards. USB-C plug did nothing, no enumeration, but the board ran happily off a separate supply. Traced it back the same way you did and found the data line was open between the chip and the socket.
Here's what worked for me. First put your meter on the small passives sitting between the chip and the connector. On some of these clone boards there's a pair of tiny series resistors in the data lines, and on mine one of them had a cracked joint you could not see without a loupe. Touching it up with fresh solder and a bit of flux fixed one board entirely. If the resistors check out, follow the trace with continuity beeps untill you find where it dies. On mine it was a via right under the connector footprint, probably cracked from the factory or damaged when the socket was hand-soldered.
Once you find the break, scrape a little of the green mask off each side with a craft knife, tin it, and lay a thin wire across. Ugly but it has survived months of plugging and unplugging. Honestly if the board was cheap I'd just do the fix, returns on these are a coin flip anyway.