Consider: a vector field $X=\nabla \phi$ on a compact, semi-simple, connected matrix Lie group $G$ where $\phi$ as a smooth scalar field on $G$ possessing only a single maxima which topologically is a point, and $\nabla$ is the gradient w.r.t to the bi-invariant metric (unique up to a constant factor which makes no difference here).

Is the following true: the set $S = \left\{ X|_g g^{-1} \ \big| \ g \in G \right\}$ must span $\mathfrak{g}(n)$ (the lie algebra of $G$)?

here $X|_g g^{-1}$ is the right translation of $X|_g$ to the identity of $G$, as we have a matrix Lie group, this composition of a tangent vector and a group element is just matrix multiplication.