When doing a line integral in the field $\vec{F}=\frac{-y}{x^2+y^2}\hat{i}+\frac{x}{x^2+y^2}\hat{j}$, the Fundamental Theorem of Line Integrals (FTOLI) almost always works because $\vec{F}$ is the gradient of $-arctan(\frac{x}{y})$. This should mean that the line integral over a closed curve is $0$, but this doesn't work when the curve encloses the singularity at $(0,0)$. Instead, the line integral over a closed curve needs to be evaluated without the FTOLI.
What I want to know is why a singularity messes up the FTOLI for a line integral around it. Since the line integral doesn't pass through the singularity, why should it change anything?