I believe that's the same model I installed in my 2018 S model but it's been 4 years so I don't remember much. I didn't attach the clutch switch wire because I'm lazy but it has always worked perfectly. When the key is off there is no light on in the switch.
I remember the instructions being written for a professional installer, not a do it yourselfer. Had to do some searching on how to remove the dash panels. Depending on where you bought it the seller may provide installation info. Once you get it working you'll like it.
I believe the VSS data on the Versa comes from the OBD plug rather than having to splice into the VSS wire somewhere. As you suggest, the problem may just be the wrong wire on the brake light switch.