I would indeed prefer the odometer based solution to the GPS
but it's not perfect
(and you are currently exempted from the gax tax model if you have a electric vehicle on private roads / tracks, I doubt too many are affected by this though)
It's not really an edge case. Everyone drives on paved private property (parking lots and the like). A flat 2% mileage reduction per year is probably enough to cover this for the vast majority of people. So if someone drives 10,000 miles a year, you'd charge them for 9800 miles with the assumption that they drove about 200 miles on private property that year.
Also, those aren’t exempted under the current gas tax model anyway, so why bother?