The fundamental fiber sequence in étale homotopy theory

Let $k$ be a field with separable closure $\bar{k}\supset k$, and let $X$ be a qcqs $k$-scheme. We use the theory of profinite Galois categories developed by Barwick-Glasman-Haine to provide a quick conceptual proof that the sequences \begin{equation*} Π_{<\infty}^{$\acute{\mathrm{e}}\mathrm{t}$}(X_{\bar{k}}) \to Π_{<\infty}^{$\acute{\mathrm{e}}\mathrm{t}$}(X) \to $\mathrm{B}\mathrm{G}\mathrm{a}\mathrm{l}$(\bar{k}/k) \qquad $\text{and}$ \qquad \widehatΠ{}{\infty}^{$\acute{\mathrm{e}}\mathrm{t}$}(X{\bar{k}}) \to \widehatΠ{}_{\infty}^{$\acute{\mathrm{e}}\mathrm{t}$}(X) \to $\mathrm{B}\mathrm{G}\mathrm{a}\mathrm{l}$(\bar{k}/k) \end{equation*} of protruncated and profinite étale homotopy types are fiber sequences. This gives a common conceptual reason for the following two phenomena: first, the higher étale homotopy groups of $X$ and the geometric fiber $X_{\bar{k}}$ are isomorphic, and second, if $X_{\bar{k}}$ is connected, then the sequence of profinite étale fundamental groups $1\to \widehat{\pi }{}_1^{\acute{\mathrm{e}}\mathrm{t}}(X_{\bar{k}})\to \widehat{\pi }{}_1^{\acute{\mathrm{e}}\mathrm{t}}(X)\to \mathrm{G}\mathrm{a}\mathrm{l}(\bar{k}\mathrm{/}k)\to 1$ is exact. It also proves the analogous results for the `groupe fondamental élargi’ of SGA3.