Yes, this is possible. Since finite groups are algebraic groups, pro-finite groups are pro-algebraic groups. So one can recover $\pi_1$ in exactly the way you say from the category of algebraic representations of $\pi_1$, i.e. finite-dimensional representations that are continuous for the discrete topology of $k$. (These all factor through a finite group).

In geometric terms, these are locally-constant finite-rank sheaves of $k$-vector spaces in the étale topology, if $X$ is normal - if $X$ is not normal there may be locally constant sheaves where the $\pi_1$-action is not continuous.

For these purposes, the choice of field $k$ doesn't matter very much.

But it's usually more interesting to study a different construction, where you consider finite-dimensional representations of $\pi_1$ that are continuous for the $\ell$-adic topology on $k$ for some $\ell$-adic field $k$. These form a Tannakian category, and the resulting pro-algebraic group is usually not pro-finite, and thus much larger than $\pi_1$. But its component group recovers $\pi_1$, basically because the representations of the component group are the category we discussed above.

Geometrically, this is (at least for $X$ normal) the category of lisse $\ell$-adic sheaves defined the usual (slightly complicated) way.

For this construction, the field matters a lot - it must be $\ell$-adic, and we get very different groups for different primes $\ell$ (though there are known to be some relationships between their representations, at least for schemes over finite fields).

definitionof the étale fundamental group an answer? I.e. the étale fundamental group is “the thing that acts on geometric fibers of finite étale covers,” exactly analogous to the Tannakian setting. $\endgroup$