Why do we use big O rather than $\Omega$ when discussing best case runtime?
The symbols $O$, $\Omega$ and $\Theta$ mean at most, at least and exactly, respectively. What these bounds are about, is an entirely separate issue. They can be about best-case or worst-case performance, but you can use these symbols throughout mathematics. To describe a function, one of the things you can do is to give $O$ and $\Omega$ upper and lower bounds, independent of whether your function is related to computer science.

For example, to describe an complicated algorithm (or language) for which you don't have precise bounds yet, but for which you do have rough preliminary bounds, you can say: the best-case of this algorithm runs in $\Omega(n^2)$ and $O(n^3)$ (meaning no better than $n^2$ but no worse than $n^3$) and the worst-case runs in $\Omega(2^{\sqrt{n}})$ and $O(2^{n^{2}})$. This means: I don't know what the worst-case running time is exactly, but my analysis thus far puts it between $2^{\sqrt{n}}$ and $2^{n^{2}}$. Later, it may turn out that the worst-case running time was in fact $\Theta(2^{n})$, and then your $O(2^{n^{2}})$ upper bound, while correct, was not strict.