Given a function which we would want to solve for some $U_\delta(x_0)$. If the IFT does not hold, for example $\frac{\partial F}{\partial x}$ is not invertible at our point of interest, how could we show that the function is still locally solvable for $x$, or show that it is in fact not solvable. Is there any general way to approach such problems? For a specific example, you may want to look at this question.
Maybe I am missing some intution about the IFT for which any answer or link to a question already answered in the forum is highly appreciated.
I think that this could help you. There must be cases where the implicit function theorem fails, but you can still have solutions; But that link provides you an example where the IFT fails, but still have no solutions in any neighbourhood. I think that it depends on your function $F$.