I tried flashing the GD32F350 using the ST-Link probe, and it failed with the error:
Error: init mode failed (unable to connect to the target)
This is the same error I get when trying to program a Blue Pill. So either there’s something wrong with both the GD32F350 and the Blue Pill, or else I’m doing something wrong. (Although somehow my bad luck did not extend to the Discovery F411, which worked okay.)