Is the map on etale fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

This is an expansion of my comments above. You do not need resolution of singularities or SGA 4. The key step is "elimination of ramification" or "Abhyankar's Lemma". This is proved in Append. 1 of Exposé XIII of SGA 1. Here is the link in the Stacks Project.

Abhankar's Lemma, Stacks Project Tag 0BRM
http://stacks.math.columbia.edu/tag/0BRM

Here is the setup for Abhyankar's Lemma. Let $A$ be a DVR that contains a characteristic $0$ field, let $A\subset B$ be an injective, local homomorphism of DVRs with ramification index $e$, i.e., $\mathfrak{m}_B^{e+1}\subset \mathfrak{m}_AB \subset \mathfrak{m}_B^e$, let $K_1/\text{Frac}(A)$ be a finite field extension, let $L_1/\text{Frac}(B)$ be a compositum of $K_1/\text{Frac}(A)$ and $\text{Frac}(B)/\text{Frac}(A)$, let $A\subset A_1$, resp. $B\subset B_1$, be the integral closure of $A$ in $K_1$, resp. of $B$ in $L_1$. Let $\mathfrak{n}_A\subset A_1$ be a maximal ideal that contains $\mathfrak{m}_A A_1$, and let $\mathfrak{n}_B\subset B_1$ be any maximal ideal that contains $\mathfrak{m}_B B_1 + \mathfrak{n}_A B_1$.

Abhyankar's Lemma. if the ramification index $e$ of $A\subset B$ divides the ramification index of $A\to (A_1)_{\mathfrak{n}_A}$, then $(A_1)_{\mathfrak{n}_A}\subset (B_1)_{\mathfrak{n}_B}$ is formally smooth, i.e., the ramification index equals $1$.

Nota bene. In characteristic $0$ this follows easily from the Cohen Structure Theorem, etc. In positive characteristic, Abhyankar's lemma says more, because (1) the tensor product $\text{Frac}(B)\otimes_{\text{Frac}(A)} K_1$ may be nonreduced, and (2) under the assumption that the the residue field extension $A/\mathfrak{m}_A \to B/\mathfrak{m}_B$ is separable, we need to also prove that the residue field extension of $(A_1)_{\mathfrak{n}_A} \to (B_1)_{\mathfrak{n}_B}$ is separable. This requires a further assumption that $e$ is prime to $p$. When $e$ is not prime to $p$, Abhyankar's Lemma has counterexamples, but the result that sometimes does the job is Krasner's Lemma, Stacks Project Tag 0BU9: http://stacks.math.columbia.edu/tag/0BU9

Let $K$ be a field (not necessarily characteristic $0$), let $X_K$ be a projective, connected $K$-scheme, and let $x_0\in X_K$ be a $K$-rational point. For every $K$-scheme $T$, an étale cover of $X_T$ trivialized over $x_0$ is a finite, étale morphism $g_T:Z_T\to X_T$ of some degree $d$ together with an ordered $n$-tuple of $T$-morphisms $(s_i:T\to Z_T)_{i=1,\dots,d}$ such that the union of the images of the sections $s_i$ equals $Z_T\times_{X_K} \text{Spec}\kappa(x_0)$.

Rigidity in the Projective Case. For every $K$-scheme $T$, every étale cover of $X_T$ trivialized over $x_0$ is isomorphic to the base change of an étale cover of $X_K$ trivialized over $x_0$, and that trivialized étale cover is unique up to unique isomorphism.

This is, essentially, proved in Section 1 of Exposé X of SGA 1. The key point is rigidity of trivialized étale covers in the proper case.

Descent for Affine Curves. Now assume further that $K$ is a characteristic $0$ field that contains all roots of unity. Assume that $X_K$ is a smooth, projective, connected curve over $K$. Let $Y_K\subset X_K$ be a proper closed subset, i.e., a finite set of closed points. Denote $X_K\setminus Y_K$ by $U_K$, and assume that the $K$-point $x_0$ is contained in $U_K$.

Fix an integer $d$. Let $f_K:\widetilde{X}_K\to X_K$ be a finite surjective morphism of some degree $n$ with $\widetilde{X}_K$ a smooth, projective curve such that (i) $f_K^{*}(x_0)$ is a set of $n$ distinct $K$-rational points of $\widetilde{X}_K,$ and such that for every closed point $y\in Y_K,$ for every closed point $\widetilde{y}\in \widetilde{X}_K$ with $f(\widetilde{y})=y$, the ramification index of $\mathcal{O}_{X_K,y}\to \mathcal{O}_{\widetilde{X}_K,\widetilde{y}}$ is divisible by $e$ for every $e\leq d$. For instance, begin with a finite morphism $g:X_K\to \mathbb{P}^1_K$ that is smooth at every point of $Y_K$, such that $g(x_0)$ equals $[1,1]$, and such that $Y_K\subset f^{-1}(\underline{0}+\underline{\infty})$, and define $\widetilde{X}_K$ to be the normalization of the fiber product of $g$ and the morphism $\mathbb{P}^1_K\to \mathbb{P}^1_K$ by $[z_0,z_1]\mapsto [z_0^{d!},z_1^{d!}]$.

For a field extension $L/K$, the ramification hypothesis on $f_L:\widetilde{X}_L\to X_L$ over $Y_L$ is still valid. For every finite surjective morphism $W_L\to X_L$ of degree $d$, for every closed point $w$ of $W_L$ that maps to $Y_L$, the ramification index $e$ at $w$ divides $d!$. Thus, by Abhyankar's Lemma, the normalization $\widetilde{W}_L$ of $W_L\times_{X_L} \widetilde{X}_L$ is étale over $\widetilde{X}_L$ at every closed point lying over $Y_L$. Thus, if $W_L\to X_L$ is assumed to be étale away from $Y_L$ then $\widetilde{W}_L\to \widetilde{X}_L$ is everywhere étale of finite degree $d$. Assume further that the fiber of $W_L$ over $x_0$ is a set of $d$ distinct $L$-rational points. Then also the fiber of $\widetilde{W}_L$ is a set of distinct $L$-rational points. By the projective descent result above, there exists a finite, étale morphism $\widetilde{W}_K\to \widetilde{X}_K$ whose fiber over $x_0$ is a set of distinct $K$-rational points and whose base change equals $\widetilde{W}_L$. Thus, $W_L$ is an intermediate extension of the base change to $L$ of the extension $\widetilde{W}_K\to X_K$. In particular, for the fiber $\widetilde{W}_K \times_{X_K} \text{Spec}\kappa(x_0)$ over $x_0$, the degree $n$ morphism $\widetilde{W}_L \to W_L$ defines a partition $\Pi$ of the fiber into $d$ subsets of size $n$.

Descent for $W_L$. There exists a unique irreducible component $W_K$ of the relative Hilbert scheme $\text{Hilb}^n_{\widetilde{W}_K/X_K} \to X_K$ whose fiber over $x_0$ parameterizes the partition $\Pi$ above. The base change of $W_K\to X_K$ to $L$ is isomorphic to $W_L\to X_L$.

In conclusion, for the open subset $U_K=X_K\setminus Y_K$, for every finite étale morphism $V_L\to U_L$, there exists a finite étale morphism $V_K\to U_K$ whose base change to $L$ equals $V_L\to U_L$.

Descent in Arbitrary Dimension. Now let $k$ be a characteristic $0$ field containing all roots of unity, and let $U_k$ be a normal, quasi-projective variety of dimension $r\geq 1$ together with a specified $k$-rational point $u_0$ in the smooth locus. I claim that there exists a blowing up $\nu_k:U'_k\to U_k$ at finitely many points including $u_0$ such that $U'_k$ is normal, and there exists a flat morphism $\pi:U'_k\to \mathbb{P}^{r-1}_k$ such that the exceptionial divisor $E_0$ over $u_0$ is the image of a rational section of $\pi$. The easiest way to get this is to embed $U_k$ into a projective space $\mathbb{P}^{r+s}_k$, choose a linear $\mathbb{P}^s_k\subset \mathbb{P}^{r+s}_k$ that intersects $U'_k$ transversally in finitely many points including $u_0$, and then let $\pi$ be the restriction to $U_k$ of the linear projection away from $\mathbb{P}^s_k$. Now let $K$ be the fraction field $k(\mathbb{P}^{r-1}_k)$ of $\mathbb{P}^{r-1}_k$, and let $U_K$ be the generic fiber of $\pi$. Let $x_0$ be the $K$-rational point corresponding to the exceptional divisor $E_0$.

For every field extension $\ell/k$, for every finite étale morphism $V_\ell\to U_\ell$ whose fiber over $u_0$ is a set of distinct $\ell$-rational points, the fiber product with $\nu_\ell:U'_\ell\to U_\ell$ is a finite étale morphism to $U'_\ell$ that is trivialized over $E_0$. Thus, setting $L=k(\mathbb{P}^{r-1}_\ell)$, we obtain a finite étale morphism $V_L\to U_L$ whose fiber over $x_0$ is a set of distinct $L$-rational points. Applying the curve case, this descends the generic fiber of $V_\ell\to U_\ell$ to a variety $V_K$ over $K=k(\mathbb{P}^{r-1}_k)$. Finally, we can construct $V_k\to U_k$ as the integral closure of $U_k$ in the fraction field of the $V_K$.

I think one can do without resolution of singularities if one has van Kampen.

Recall that an elementary fibration [SGA 4 III Exp. XI] is a morphism of schemes $f:X\to S$ admitting a factorization $f=\bar f\circ j$ into an open immersion $j:X\to \bar X$ and a smooth projective morphism $\bar f:\bar X\to S$ whose geometric fibers are connected curves, and such that if $Y=\bar X\setminus X$, then $\bar f|_Y : Y\to S$ is finite \'etale surjective. Thus $f$ is a `fibration in affine curves,' and it is not difficult to show that (for $S$ connected) there is a short exact sequence $$ 1\to \pi_1(X_{\bar s}, \bar x) \to \pi_1(X, \bar x)\to \pi_1(S, \bar s) \to 1. $$ Here $\bar x$ is a geometric point of $X$, $\bar s = f\circ \bar x$.

Now suppose $X$ is smooth over $k$. A good neighborhood of a point $x\in X(k)$ is an open subset $U\subseteq X$ containing $x$ and admitting a chain of elementary fibrations $$ U = U_{\dim X} \to \ldots \to U_1 \to U_0 = {\rm Spec}\, k. $$ For a good neighborhood $U$, $U_L$ is a good neighborhood over $L$, and the natural maps $\pi_1(U_L)\to \pi_1(U)$ are isomorphisms. This is proved by induction on the dimension, using the above short exact sequence and the curve case.

Artin proved in op.cit. that $X$ admits an open covering by good neighborhoods. By van Kampen, this allows one to exhibit $\pi_1(X)$ as a colimit of a finite diagram of $\pi_1(U)$ for some good neighborhoods $U$. Base changing to $L$, we deduce from the fact that $\pi_1(U_L)\cong \pi_1(U)$ that $\pi_1(X_L)\cong \pi_1(X)$.

I ended up writing a proof of the isomorphism $\pi_1(U_L)\rightarrow \pi_1(U)$ in my question above, including a characteristic p version. It is essentially a re-organization of Jason Starr's proof above, together with some suggestions by Brian Conrad. You can find the write up at https://arxiv.org/abs/2005.09690