Set $p = \text{char}(K)$. By Lemma \ref{lemma-compare-cohomology-point}, Theorem \ref{theorem-brauer-delta}, and Example \ref{example-sheaves-point} the proposition is equivalent to showing that if  $H^2(\Spec(K'),\mathbf{G}_m|_{\Spec(K')_\etale}) = 0$ for all finite extensions $K'/K$ then: \begin{itemize} \item $H^q(\Spec(K),\mathbf{G}_m|_{\Spec(K)_\etale}) = 0$ for all $q \geq 1$, and \item $H^q(\Spec(K),\mathcal{F}) = 0$ for any torsion sheaf $\mathcal{F}$ and any $q \geq 2$. \end{itemize} We prove the second part first. Since $\mathcal{F}$ is a torsion sheaf, we may use the  $\ell$-primary decomposition as well as the compatibility of cohomology with colimits (i.e, direct sums, see Theorem \ref{theorem-colimit}) to reduce to showing  $H^q(\Spec(K),\mathcal{F}) = 0$, $q \geq 2$ for all $\ell$-power torsion sheaves for every prime $\ell$. This  allows us to analyze each prime individually. \medskip\noindent Suppose that $\ell \neq p$. For any extension $K'/K$ consider the Kummer sequence (Lemma \ref{lemma-kummer-sequence}) $$ 0 \to \mu_{\ell, \Spec{K'}} \to \mathbf{G}_{m, \Spec{K'}} \xrightarrow{(\cdot)^{\ell}} \mathbf{G}_{m, \Spec{K'}} \to 0 $$ Since $H^q(\Spec{K'},\mathbf{G}_m|_{\Spec(K')_\etale}) = 0$ for $q = 2$ by assumption  and for $q = 1$ by Theorem \ref{theorem-picard-group} combined with $\Pic(K) = (0)$. Thus, by the long-exact cohomology  sequence we may conclude that $H^2(\Spec{K'}, \mu_\ell) = 0$ for  any separable $K'/K$. Now let $H$ be a maximal pro-$\ell$ subgroup of the absolute Galois group of $K$ and let $L$ be the  corresponding extension. We can write $L$ as the colimit of finite extensions,  applying Theorem \ref{theorem-colimit} to this colimit we see that $H^2(\Spec(L), \mu_\ell) = 0$. Now $\mu_\ell$ must be the constant sheaf. If it weren't, that would imply there exists a Galois extension of degree relatively prime to  $\ell$ of $L$ which is not true by definition of $L$ (namely, the extension one gets by adjoining the  $\ell$th roots of unity to $L$). Hence, via Lemma \ref{lemma-reduce-to-l-group-higher}, we conclude the result for $\ell \neq p$.  \medskip\noindent Now suppose that $\ell = p$. We consider the Artin-Schrier exact sequence (Section \ref{section-artin-schreier}) $$ 0 \longrightarrow \underline{\mathbf{Z}/p\mathbf{Z}}_{\Spec{K}} \longrightarrow \mathbf{G}_{a, \Spec{K}} \xrightarrow{F-1} \mathbf{G}_{a, \Spec{K}}  \longrightarrow 0 $$ where $F - 1$ is the map $x \mapsto x^p - x$. Then note that the higher  Cohomology of $\mathbf{G}_{a, \Spec{K}}$ vanishes, by Remark \ref{remark-special-case-fpqc-cohomology-quasi-coherent} and the  vanishing of the higher cohomology of the structure sheaf of an affine scheme (Cohomology of Schemes, Lemma \ref{coherent-lemma-quasi-coherent-affine-cohomology-zero}). Note this can be applied to any field of  characteristic $p$. In particular, we can apply it to the field extension $L$  defined by a maximal pro-$p$ subgroup $H$. This allows us to conclude  $H^n(\Spec{L}, \underline{\mathbf{Z}/p\mathbf{Z}}_{\Spec{L}}) = 0$ for $n \geq 2$, from which the result follows for $\ell = p$, by Lemma \ref{lemma-reduce-to-l-group-higher}. \medskip\noindent To finish the proof we still have to show that $H^q(\text{Gal}(K^{sep}/K), (K^{sep})^*) = 0$ for all $q \geq 1$. Set $G = \text{Gal}(K^{sep}/K)$ and set $M = (K^{sep})^*$ viewed as a $G$-module. We have already shown (above) that $H^1(G, M) = 0$ and $H^2(G, M) = 0$. Consider the exact sequence $$ 0 \to A \to M \to M \otimes \mathbf{Q} \to B \to 0 $$ of $G$-modules. By the above we have $H^i(G, A) = 0$ and $H^i(G, B) = 0$ for $i > 1$ since $A$ and $B$ are torsion $G$-modules. By Lemma \ref{lemma-profinite-group-cohomology-torsion} we have $H^i(G, M \otimes \mathbf{Q}) = 0$ for $i > 0$. It is a pleasant exercise to see that this implies that $H^i(G, M) = 0$ also for $i \geq 3$.