The answers in this post say otherwise, but I'm not sure if they are commenting on the set $\{ \sin n \pi x\}$ itself or its span. I was wondering if the following argument was valid:
We know that Fourier series are dense in $L^2[-1, 1]$, and that in particular, for $f$ in $L^2$, its Fourier series converges to $f$ (with respect to the $L^2$ norm). We can extend any $g \in L^2[0,1]$ to an odd function $h$ on $[-1, 1]$, and the Fourier series of an odd function is a sine series, so $h(x)\sim \sum \hat{h}(n) \sin(n \pi x)$. Then restricting $h$ back to $[0, 1]$, we still have $h(x)=g(x)\sim \sum \hat{h}(n) \sin(n \pi x)$ in the $L^2$ metric.
Your argument is correct and the span is dense. The question in your link is about denseness of the sequence itself, not its span.