A weakness in the later parts of all these sources is in the treatment of algebraic functions, and of solutions to ordinary differential equations with regular singular points. And some so-called 'special functions' such as the gamma function deserve fuller treatment, especially with an eye to applications in other parts of mathematics.
And the ultra-classical aspects of elliptic functions and modular forms deserve a quick and decisive treatment in this context, since all the crucial techniques at this level do really belong to 'complex analysis', as opposed to 'number theory' or anything else.
And the connection between the Riemann Hypothesis on the zeta function and error terms in the Prime Number Theorem is hard to ignore, motivating as it did a great deal of work in classical complex analysis.