Gibbs–Helmholtz equation
If you read the Wikipedia page for Nernst's Heat Theorem carefully, it explains that the novelty lies not in showing (as you point out) that as $T \rightarrow 0$, $\Delta G \rightarrow \Delta H$, but rather that as   $T \rightarrow 0$, the slope of $\Delta G$ as a function of T goes to 0, that is

$$\lim_{T\rightarrow0} \left( \frac{\partial \Delta G}{\partial T} \right)_p =0$$ 

Then, since that slope is given by $-\Delta S$, it predicts the experimentally confirmed conclusion that as $T \rightarrow 0$, $\Delta S \rightarrow 0$,