Approximating convex sets
The construction you suggest does work, but it is not optimal. The reason why people use the sphere with a constant expansion is that then a $\sqrt{\epsilon}$-net is enough. That reduces the size of the coreset from $O(1/\epsilon^{d-1})$ to $O(1/\epsilon^{(d-1)/2})$. To see that replacing $\epsilon$ by $\sqrt{\epsilon}$ in your construction does not work, consider the case of a regular triangle (or simplex) where only points within distance roughly $\sqrt{\epsilon}$ of the vertices are sampled.