The point is that in general cases we cannot ensure the existence of $\phi$ such that $$ d\phi(z) =A(z) \cdot dz \quad \Leftrightarrow \quad \frac{\partial \phi(z)}{\partial z^\mu} =A_\mu(z) \quad \text{for } \forall z \quad . $$ For $\phi$ to exist, the condition $ \frac{\partial A_\nu}{\partial z_\mu} = \frac{\partial A_\mu}{\partial z_\nu} $ is needed.
Setting $ z=x-\epsilon/2 +t\epsilon $, The correct calculation goes as follows. $$ \frac{\partial}{\partial x_\mu} \int_{x-\epsilon/2}^{x+\epsilon/2} dz \cdot A(z) = \frac{\partial}{\partial x_\mu} \int_0^1A_\nu(x -\epsilon/2+t \epsilon) \, d(t \epsilon)^\nu = \epsilon^\nu \int_0^1 \frac{\partial}{\partial x_\mu} A_{\nu}(x -\epsilon/2+t \epsilon) dt \ = \epsilon^{\nu} \int_0^1 \left( \frac{\partial}{\partial x_\mu} A_\nu(x) +{\cal O}(\epsilon) \right) dt = \epsilon^\nu \frac{\partial}{\partial x_\mu} A_\nu(x) +{\cal O}(\epsilon^2) $$
I think the last line of (19.24) should be replaced by $$ + \bar{\psi}(x+\frac{\epsilon}{2})\gamma^{\mu}\gamma^{5}[-ie\epsilon^{\nu}\partial_{\mu}A_{\nu}(x) + O(\epsilon^2) ]\text{exp}\bigg[-ie \int_{x-\epsilon/2}^{x+\epsilon/2}dz\cdot A(z)\bigg] \psi(x-\frac{\epsilon}{2})\bigg} . $$ Because the authors doesn't restrict terms within order $\epsilon$ at this stage, and the <PERSON> line exists in the first and the second line of (19.24). It is a misprnt. 677
($ \epsilon^{\nu}\partial_{\mu}A_{\nu}(x) $ is correct there.)
In this case considering $$ \text{symm }\lim_{\epsilon\rightarrow0}=\left{\frac{\epsilon^{\mu}}{\epsilon^2}\right} =\infty ,\quad \text{symm }\lim_{\epsilon\rightarrow0} \Bigl{\frac{\epsilon^{\mu}\epsilon^{\nu}}{\epsilon^2}\Bigr} = \frac{g^{\mu \nu}}{d} , \tag{19.23} $$ (the first of these is misprinted in the textbook. And since the real identity of $\epsilon$ is a mystery, we may take absolute value like $\text{symm }\lim_{\epsilon\rightarrow0}=\lvert\frac{\epsilon^{\mu}}{\epsilon^2}\rvert =\infty$. Reference:here), the path of the line integral is fixed. Hence we cannot ensure the existence of $\phi$ for the reason that we cannot claim the line integral gives the same value for any path. Even if we define $\varphi(x)= \int_{x_i}^{x} dz \cdot A(z) $ for a fixed path(at least numerically calculable), $\partial_{\mu} \varphi(x)$ doesn't exist. 9