The only thing I've heard about not recalibrating for tire size is that it can cause the lights to come on during high speed turns, like exiting off ramps. I did not read that the lights stay on however, but they might.
It does sound like a speed sensor. But if you checked the sensors, and the connectors, and the wires for breaks, then I'm not sure what else it could be. Good luck and post what you find for the next guy.
2014 JK Sport, auto, 35x12.5x17, 3.73s.
I jeep therefore I am.