As Christian Remling points out, the subtleties appear for $c$ close to $1/2$. That limit can be obtained rather directly from the characteristic function of the Irwin-Hall distribution. I find the principal value integral $$P_c\equiv\mathbb{P}\left[\sum_{i=1}^n X_i \leq n \cdot c\right]=\frac{1}{2}-\frac{1}{2\pi}\int_{-\infty}^\infty dt\,\frac{1}{(it)^{n+1}}\left(e^{it(1-c)}-e^{-ict}\right)^n$$ or, if we write $c=1/2+\delta$, $$P_{1/2+\delta}=\frac{1}{2}+\frac{1}{\pi}\int_{0}^\infty dt\,\sin(2nt\delta)\,\frac{\sin^n t}{t^{n+1}}$$
For small $\delta$ this evaluates to
$$P_{1/2+\delta}=\tfrac{1}{2}+n\delta+{\cal O}(\delta^3)$$
which is indeed an increasing function of $n$.