Since this question has not attracted any answers yet, I have substantially rewritten it in the hopes that I will make it clearer.
The first time students are exposed to Cauchy's Theorem, they are usually shielded from some of the difficulties by a rather restricted formulation which does not need any understanding of the Jordan curve theorem etc. To quote Stein/Shakarchi "at this early stage of our study, we shall make use of the device of limiting ourselves to regions whose boundaries are curves that are "toy contours".
I have recently been revising my understanding of this area. I came across Concise Complex Analysis by Sheng Gong. Gong appears to be a high-class Chinese researcher who is also interested in pedagogy. His book is unusually short (170 small pages), but also unusually concise and advanced. He also makes a point of assuming that his readers will want to go on to study functions of more than one complex variable. Unfortunately, World Scientific did not serve him well and the translation into English is not felicitous. In $\color{red}{\text{the Foreword}}$ we have:
From the point that theory of functions of one complex variable is, in the essence, the calculus in the complex number field, the fundamental theorem of calculus proves to be the Green theorem of complex form in the complex number field which promptly leads to Cauchy-Green Formula, ie Pompeiu Theorem. Now the real part and imaginary part of the function belong to $C^1$, unnecessarily asking for that the function should be holomorphic. Here the Cauchy's integral theorem and formula naturally become its result by simple inference. Then, why should the Pompeiu Theorem be first introduced instead of the Cauchy's integral theorem and formula? It is because: (A) it is just to follow a logic train as a matter of course from the point that theory of functions of one complex variable is the calculus in the complex number field. That is from complex form of the Green Theorem, the result of the inference, unconditionally, should be Pompeiu Formula, not the Cauchy's integral formula; and (B) the solution of one dimensional $\overline{\partial}$ equation could be inferred by Pompeiu Theorem, which cannot be obtained by the Cauchy integral formula. It is known that $\overline{\partial}$ question is a very important part in the modern theory of partial differential equation, and a powerful tool in the modern mathematics.
I had never heard of Pompeiu's theorem, so was somewhat foxed. [In passing, I see Wikipedia uses the name for an apparently unrelated theorem in elementary geometry.]
His definitions turn up later. In $\color{fuchsia}{\text{Chapter II}}$ we have
On p37 Theorem 1 (Cauchy-Green Formula, Pompeiu Formula) Let $U\subset\mathbb{C}$ be a bounded domain with $C^1$ boundary, ie the boundary is a smooth curve. Let $f(z)=u(x,y)+iv(x,y)\in C^1(U)$, that means $u(x,y),v(x,y)$ have first order continuous partial derivatives, then for any $z\in U$ $$f(z)=\frac{1}{2\pi i}\int_{\partial U}\frac{f(\zeta)}{\zeta-z}\ d\zeta-\frac{1}{2\pi i}\int\int_U \frac{\partial f(\zeta)}{\partial\overline{\zeta}}\frac{d\overline{\zeta}\wedge d\zeta}{\zeta-z}$$ $$=\frac{1}{2\pi i}\int_{\partial U}\frac{f(\zeta)}{\zeta-z}\ d\zeta-\frac{1}{\pi}\int\int_U \frac{\partial f(\zeta)}{\partial\overline{\zeta}}\frac{dA(s)}{\zeta-z}$$
and also
Theorem 4 (solution of one-dimension $\overline{\partial}$-equation) Let $\psi(z)\in C^1(C)$ with a compact support, ie the support is compact. Let $$u(z)=-\frac{1}{2\pi i}\int\int_C\frac{\psi(\zeta)}{\zeta-z}\ d\overline{\zeta}\wedge d\zeta$$ then $u(z)\in C^1(C)$ and $u(z)$ is the solution of the $\overline{\partial}$-equation $\frac{\partial u(z)}{\partial\overline{z}}=\psi(z)$
All of which left me little wiser.
So my questions are:
(1) Could someone explain what Gong is saying in $\color{red}{\text{the Foreword}}$. Why, for example, is the "$\overline{\partial}$ question ... a very important part in the modern theory of partial differential equation, and a powerful tool in the modern mathematics"?
(2) Has anyone any general wisdom or guidance to impart on these matters;
(3) Can anyone recommend any other books, articles, videos, MSE questions etc which might help me.