Can the GC553G2 be connected to a USB3 Type A port, or Thunderbolt?

Yes, it can.

If your computer's USB3 Type A port supports 10Gbps and you use a 10Gbps Type C to Type A cable, you can achieve 4K60 capture.
If your USB3 Type A port supports only 5Gbps, your platform can support a maximum of 4K30 capture.

By the way, GC553G2 doesn't support Thunderbolt.