USB-C on it's own doesn't offer any performance gains when it comes to data transfer. Its just a connector shape. In order to get the full SATA bandwidth (6 Gbps) you need to make sure that the cable you are using is rated for a higher bandwidth (10 Gbps/USB3.2 Gen 2).
The other USB spec (3.2 Gen 1) is capped at 5gbps and won't let you use SATA at full capacity.
OTOH, since you mentioned you were getting workable speeds via USB 2.0 (480 Mbps), chances are your USB-C cable is damaged and not even able to let 480 megabits through in a second or is crippled by some interference.
If you are using any port adapter (Like USB A to C or even USB whatever to SATA), make sure it is also able to support the expected bandwidth.
Most of the USB-C adapters and cables intended for power supply are just using USB 2.0 for data transfer and sometimes even not tested for data.
TLDR; try using a cable and/or adapters rated for 10 Gbps