@Oscar Randal-Williams: Oh, neat! I was so fixated on understanding why this should work, I wasn't looking for counterexamples. I also now understand why Hsiang's proof doesn't work without changes for $G=\mathbb{Z}_p$ and cohomology with $\mathbb{Z}$ coefficients: $S=H^*(BG;\mathbb{Z})-\{0\}$ is not a multiplicative system, since there are elements in the $0$-th gradation that will be zero divisors. But I still don't see what breaks down if I get rid of those and take $S=H^*(BG;\mathbb{Z})-H^0(BG;\mathbb{Z})\cup \{1\}$.

...Now, if $G$ is a $p$-torus and $k=\mathbb{F}_{\ell}$ for $\ell \neq p$, then the said kernel is zero, because $H^*(BG;k)$ vanishes above dimension $0$ and the said map is the identity on $H^0$. And $0$ is most certainly not in $S$. Having said that, I don't think I understand the reason why it fails.

@Oscar Randal-Williams: I understand why the proof Hsiang gives doesn't work in this case. It boils down to understanding the intersection of $S$ with the kernel of the map $H^*(BG;k) \to H^*(BH;k)$ induced by the inclusion $H \to G$, where $H \subsetneq G$. If this is non-empty for any such $H$, then the conlusion follows from a "general" localization theorem "$S^{-1}H^*_G(X) \cong S^{-1}H^*_G(X^S)$" (which is true for any compact Lie group $G$!)...