A very good question.
It's precisely becausethe results of the coriolis force are difficult to visualise in a non-rotating frame that it's useful to look at the situation in the frame rotating with the earth, even if you have to add apparent forces.
If you look at the school textbook problem of a stone twirling around on a string, it's easy to solve the problem in either a non-rotating frame, or a frame rotating with the stone, with centrifugal force added.
But when you start looking at objects moving in the rotating frame and at extra forces being applied that are dependent on displacements in the rotating frame too (e.g. the pressure gradients on the surface of the earth, which move with the surface of the earth as it rotates), it becomes almost impossible to model and solve the problem in a non-rotating reference frame. In a non-rotating reference frame, the motion of the air in a cyclone is not a simple rotation but very complex indeed.