Whether you are interested in coefficients or in the main variable, a complex power series is a complex power series, and you use the theory of complex function

Whether you are interested in coefficients or in the main variable, a complex power series is a complex power series, and you use the theory of complex functions. You don't just take derivatives of generating functions, you use the full package, including for example convergence criteria of infinite series - that's where the power of this approach comes from in the end, that large parts of the powerful apparatus of complex analysis can be used for solving discrete problems. I don't really see any significance in the fact that formal derivatives can be defined as discrete operations, especially this only works if the powers are natural numbers, which is not the case in general. All this is tangential in the end, there are certainly some specialized areas of CS that require less calculus than others, in general though you encounter calculus a lot in almost all areas of science and engineering, and the university is supposed to give you a broad foundation for your future career, so it should teach calculus to prospective scientists and engineers.