Results

The experiment didn’t work. Although it wasn’t really an experiment more of a demonstration but on either count it failed.
A ‘demonstration’ because it ‘must’ work if carried out properly.
3 dimensional Euclidean geometry is part of the way that the world is, at least on the level of this exercise. If I had properly cut the corners in my journey home there must be a clear difference between the journey there and the journey back but there wasn’t, exactly the same 18.2 miles on the mileometer. Since the instrument only measures to one decimal place either the affect over 18 miles is less than 1/10th of a mile or I wasn’t extravagant enough in cutting the corners, probably a bit of both.
Somebody following me must have been extremely puzzled as it was because I didnt drive fast, just kept to around 40mph but on coming to a left hand bend would hug the inside line like a geriatric formula 1 driver and when encountering a right hand bend, if I couldn't safely cross onto the opposite side of the road, would put my offside wheels onto the central white line and pass the thundering Juggernaut coming in the other direction like some bizarre parody of a Red Arrows 'crossover'.