Throughout this post, we fix a field $k$ (of characteristic zero if makes our lives easier :)) and $X$ an arbitrary variety (scheme) over $k$. I understand what's the pathology/discrepancy, behind the theory of covering spaces for a fixed $X$ in algebraic geometry, and why we come up with the particular "étale" setup to mimic algebraic topology (into a certain extend of course). We cannot even define the notion of fundamental group $\pi_1(X,x)$, in a proper pointed (topological) category; so, we construct the so-called finite étale morphisms and exploit the isomorphism $\pi_1(X,x) \cong \text{Aut}(\tilde{X})$, to obtain the algebraic (étale) fundamental group.
Now my question has to do with the intuition behind the idea of geometric points. While in the category of topological spaces, we can easily think of a singleton $x \in X$ and form the pointed space $(X,x)$ (hence $\pi_1(X,x)$ makes sense) the same isn't true in the category of schemes, where additional (algebraic) data is considered. So if I understand correctly, the idea of choosing a point here, becomes equivalent with the choice of a morphism $\overline{x} \to X$ (which in turn tries somehow to mimic as well the idea of a loop lifting in the pure topological setup??). The latter though, we don't want to be arbitrary, but due to J.S. Milne - "Lectures on Étale Cohmology" the chosen geometric point, should be coming from the choice of a morphism of schemes $\text{Spec}(L) \to X$, where $L$ ($L/k$ is a finite extension), some separably algebraically closed field. So why do we need these assumptions for $L$? Why there is no other "canonical" way to choose a point in $X$?
P.S. Any reference which deals with this question is welcomed. Thanks!