Tested it on a Windows 10 notebook with USB-C cable and Steam / Garfield Kart: Works.
However, I could not determine whether the NiMh batteries are charged during the cable connection. At least the controller on the cable also works with removed batteries, so it is supplied with power from the cable.

Does it come with a long enough USB C cable to play with?

So this controller definitely does not run with a cable. It looks the same as my old one, has a socket in the same place, but the part is sealed. It is also not mentioned anywhere that you can plug in a cable.