Actually, your assessment of Mode 1 isn't completely correct
Mode 1 is like standing still by the race track, and panning to follow the subject. So depending on the angle/direction of the car's movement and the point of focus, some parts may be sharper than others.
Mode 3 is like having a camera mounted to the car with a robotic arm where the car never moves relative to the camera. The car will never be blurry.
Mode 2 would be between both. (I guess it would be like taking a picture of a car while riding in another one next to it or something lol)
I like the look of Mode 1. I find it gives realistic results where the rear end of the car might appear blurry sometimes. Using 1/125, 1/250 or 1/60 in some cases with an aperture of f/45 gives a really nice motion blur to the background. Professional photographers strive for the look of Mode 3 and sometimes resort to post-processing to obtain it.