My car GPS uses wheel rotation to improve the accuracy when you are getting close to places it wants you to make a turn. It says something like "in 50 feet, turn left" and counts tire rotations from there to where you turn.

 

Yes, increasing the sidewall 1/2 inch is likely possible. I did that on my 2004 VW Phaeton. I put on 19-inch Bentley tires and wheels. The tires were 3/4 inch "taller" than the spec for 18-inch wheels/tires that came stock. Since most speedometers read a little high to help you avoid speeding, the taller tires actually corrected that overage. It didn't seem to bother the navigation either. But of course, a VW is not an XJ.

 

I think they read high to avoid under-reading when the tyres wear down - in the UK, regulations insist they never show lower than true speed, but can over-read by up to 10%. Manufacturers are trying to keep themselves legal, not their customers!

 

Because the gps counts the number of wheel rotations when it gets close to a decision waypoint, like a turn off or stop sign. It improves accuracy over what the satellites can offer.