While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one

While what you say is true, I could never intuitively grasp that properties of analytic functions. Like I could read and understand the proofs, as in follow one step to the next, but I could never succinctly describe, intuitively, why one should expect the proofs to hold. Even the most fundamental concepts in complex analysis are more like facts rather than logical deductions (to me).