The problem isn't that $S_{k + 1/2}(N)$ is zero if $4 \nmid N$; it's that the space is not defined if $4 \nmid N$.
In order to make sense of what a "half-integer weight form of level $N$" means, you need to have a consistent way of choosing square roots of $c\tau + d$, for all $\tau$ in the upper half-plane and $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in \Gamma$. There is no sensible way of doing this for all $\Gamma$, and that's why you need the $4\mid N$ condition.
This "choice of square roots" has a highbrow interpretation in terms of lifting $\Gamma$ to a subgroup of the metaplectic group, a double covering of $\operatorname{SL}_2(\mathbf{R})$. See my answer to the following Math.SE question: https://math.stackexchange.com/questions/2802562.
