Going from position to velocity to acceleration makes sense. But suddenly acceleration to jerk is hard to grasp. Why is that?
Lots of natural systems are well defined using second order differential equations. To go beyond second order requires "interesting" systems, which are more dynamic than the garden variety. Since we see more of these systems that are easily captured with second order differential equations, our brains have developed a better intuition of them.
I am not convinced that jerk is hard to grasp. Anyone standing 4m above the ground having to jump will intuitively understand how jerk differs as they look down at the soft foam and the surrounding concrete. But, you do make an important point that things which can't be easily sensed, like snap, crackle and pop do become increasingly difficult to grasp....
Problems with a variable acceleration are usually more difficult to solve and rarely come up in a basic physics course. The most familiar jerk occurs when you are stopping for a red light. You decelerate until the velocity is zero, and then the deceleration suddenly stops.