Reference request: heuristic for analyticity of holomorphic functions

Question

I remember reading somewhere an intuitive non-rigorous heuristic of the analyticity of holomorphic functions, that involved showing that the gauss mean value property is approximately true for small r (by writing $$f (z + re^{it}) = f (z) + f ' (z) (re^{it}) + o (re^{it})$$ and showing that the integral of $e^{it}$ vanishes), and then somehow iterating it to generate higher order terms. I lost the source, but does anyone know if it's possible to finish the argument? I have been dying for an intuition behind this property that doesn't involve things like cauchy's integral formula, since that hides what is truly going on in a fundamental level. Alternatively, does anyone know any other heuristic for this fact (besides just restating the power of holomorphicity with things like CR equations)?