If we know, with the help of Jordan curve theorem and Goursat's lemma, that the Cauchy integral theorem holds for a simple closed polygon, and we also know that the integral of a continuous complex valued function over a closed curve can be approximated with an integral over a closed polygon, then how can we infer the general case of the Cauchy integral theorem?
Cauchy integral theorem. If a function $f$ is holomorphic inside a Jordan curve $J\subset \mathbb{C}$, then $\int_C f \space dz=0$ for every closed curve $C$ inside the curve $J$.
Surely it's easy if we draw a closed polygon with only a few self-intersections and add and cancel lines in a suitable manner, but I haven't found a rigorous (inductive algorithm) proof for the general case. [The proof that deals with null-homotopic curves in a simply connected domain is not the question, even though the result is equivalent.]
You don't seem to have a proper definition of a simply connected domain. What are points 'inside' a simple closed curve?. This is highly complicated question in topology and you cannot define a simply connected domain the way you have done. You are seeking a rigorous proof of a result when the concept of simply connected domain is not defined precisely. I suggest you study a standard text on Complex Analysis like the one's by John Conway Walter Rudin.