I think they are moving too fast with their diagnosis.

First, if you drive the car, does the speedometer work? IF it does not, it's the RCM or a fuse on the RCM. If the speedometer does work, I'd replace the battery first, then do the 'all the way to the left and right" thing again.

If you still cant get the lights to go out, then only then would I change the switch in the brake controller.