Because you're obviously doing it wrong.
And you're doing it wrong, because you think wrong is right and right is wrong, therefore wrong must be better than right.
I just ride the bike the same way that I'd run the course, taking the shortest route possible. No, I don't stare at the ground and I don't look at the count, which might explain why you're wobbling around like you do with the wheel. I take care that the speed of the bicycle is reasonable to maintain the integrity of the wheel with the line that I'm traveling and the ground. That means that I won't be keeping the same speed all the time, unless the course is exactly the same all the time, but that's usually not the case, even on the roads.
If you ride the course twice with a Jones Counter on a bike, and you ride the course the shortest distance each time, then you should get the exact same measurement, which should not be off more than a few inches each time. For these to be accurate, the bicycle needs to be calibrated both before, and after, the measurements. This is to verify that no significant change has occurred to the pressure of the tires, or anything else that would affect the calibration of the bike.
Walking or jogging with a wheel is not accurate because of the wobbling of the wheel, and there there is no weight on the wheel. When you are walking or jogging, your feet are not on the ground all the time, and that affects the motion of the wheel.