I don't actually see how to deduce the version of the classical SvKT on a set of base points $A$ directly from Lurie's version. It seems that to apply Lurie's theorem, we would need the stronger hypothesis that $A$ meets every path component of every *finite* intersection of open sets in $\mathcal{U}$ (which is reasonable, since the conclusion of Lurie's theorem is a stronger statement about fundamental $\infty$-groupoids rather than just fundamental 1-groupoids). But I think we can adapt his proof to derive the classical version.

Here's the first thing I'm going to try to prove: let $\chi:C\to \mathcal{O}(X)$ be a functor, with $C$ a small category and $\mathcal{O}(X)$ the poset of opens in $X$. For each $x\in X$, define $C_x$, as Lurie does, to be the full subcategory of $C$ spanned by those objects $c$ with $x\in\chi(c)$. Assume that for every $x$, the nerve of $C_x$ is *simply connected*. Then we have $\Pi_1(X) \cong \mathrm{colim}_{c\in C}\; \Pi_1(\chi(c))$, the colimit being a weak 2-colimit of groupoids.

Mimicking Lurie's argument in the 1-truncated case, we have a (pseudo 2-)functor $F:\mathrm{Gpd}^{\mathcal{O}(X)^{\mathrm{op}}} \to \mathrm{Gpd}$ defined by Kan extension from the functor $\Pi_1 : \mathcal{O}(X) \to \mathrm{Gpd}$. The (2,1)-topos $\mathrm{Sh}_{(2,1)}(X)$ is the (bicategorical) localization of $\mathrm{Gpd}^{\mathcal{O}(X)^{\mathrm{op}}}$ at the covering sieves, and Lurie's A.3.2 shows that $F$ inverts these covering sieves and hence factors through $\mathrm{Sh}_{(2,1)}(X)$. In particular, this induced functor $F:\mathrm{Sh}_{(2,1)}(X) \to \mathrm{Gpd}$ preserves (bicategorical) colimits.

Thus, it suffices to show that our functor $\chi:C\to \mathcal{O}(X)$ has colimit $X$ (the terminal object) when composed with the Yoneda embedding into $\mathrm{Sh}_{(2,1)}(X)$. And since $\mathrm{Sh}_{(2,1)}(X)$ has enough points (being sheaves on a topological space), it suffices to check this on all stalks. (At a finite categorical dimension, there is no hyper-incompleteness to worry about.) But at the stalk over $x\in X$, the $C$-diagram is trivial at those $c\in C_x$ and empty at the others, so its colimit is simply the groupoid reflection of $C_x$, which was assumed to be terminal (this is equivalent to the nerve of $C_x$ being simply connected).

This completes the proof of the 1-groupoidal version of Lurie's theorem. Now let's deduce a more classical statement. Let $X$ be our space and $\mathcal{U}$ an open cover of it. Define $C$ to be the category of 1-, 2-, or 3-fold intersections of open sets in $\mathcal{U}$, whose morphisms are the canonical inclusions from an $(n+k)$-fold intersection to an $n$-fold intersection, and let $\chi$ be the obvious functor.

For any $x\in X$, the category $C_x$ is obviously nonempty (because $\mathcal{U}$ covers $X$) and connected (because if $x\in U$ and $x\in V$, then $x\in U\cap V$). It is not much harder to see that it is simply connected: any two parallel zigzags of inclusions can be made equal by passing through at most triple intersections. Thus, we have

$$\Pi_1(X) \cong \mathrm{colim}_{c\in C} \; \Pi_1(\chi(c)).$$

Now let $A\subseteq X$ be a subset which meets all path components of all 1-, 2-, and 3-fold intersections of open sets in $\mathcal{U}$. Then $\Pi_1(\chi(c))$ is equivalent to its full sub-groupoid $\Pi_1(\chi(c),A)$ spanned by objects that are points of $A$, as is $\Pi_1(X)$. Since 2-dimensional colimits are invariant under equivalence of groupoids, the above statement passes to these groupoids as well.

Now the generalized SvKT of Ronnie and his coauthors amounts to asking that $\Pi_1(X)$ be the *strict* colimit of the functor $c\mapsto \Pi_1(\chi(c),A)$, in the 1-category of groupoids, so it basically suffices to show that this strict 1-colimit is also a (weak) 2-colimit. Now $C$ is a direct category, and $\mathrm{Gpd}$ is a model category with the canonical model structure (weak equivalences are equivalences, cofibrations are injective on objects), so $\mathrm{Gpd}^C$ inherits a Reedy model structure for which the adjunction

$$ \mathrm{colim} : \mathrm{Gpd}^C \;\rightleftarrows\; \mathrm{Gpd} : \Delta $$

is Quillen. It follows that the 1-colimit of a Reedy cofibrant diagram is also a 2-colimit. Unfortunately, $c\mapsto \Pi_1(\chi(c),A)$ is not Reedy cofibrant, but it is "partly" so. For instance, for any $U\in \mathcal{U}$, consider the latching object

$$ L_U = \mathrm{coeq}\left( \coprod_{V,W} \Pi_1(U\cap V\cap W,A) \;\rightrightarrows\; \coprod_{V}\Pi_1(U\cap V,A) \right) $$

Then the map $L_U \to \Pi_1(U,A)$ is injective on objects; thus our functor is at least Reedy cofibrant "at the top level". It will suffice to show that if $G\in \mathrm{Gpd}^C$ is "sufficiently Reedy cofibrant" in senses like this, then $\mathrm{colim}(G)$ can be calculated in a homotopy-invariant way.

Let $G\in \mathrm{Gpd}^C$, and enumerate the elements of $\mathcal{U}$ (perhaps transfinitely) as $(U_\alpha)_{\alpha<\lambda}$. We will define a transfinite sequence of groupoids
$$ H_0 \to H_1 \to H_2 \to \cdots $$
such that $H_\alpha$ is the colimit of $G$ restricted to the subcategory of $C$ determined by the $U_\beta$ with $\beta<\alpha$, and their pairwise and triple intersections. Of course we can take $H_0 = 0$. Now given $H_\alpha$, define $H_{\alpha+1}$ to be the pushout of $H_\alpha$ and $G(U_{\alpha})$ along
$$ K_\alpha = \mathrm{coeq}\left( \coprod_{\beta,\gamma < \alpha} G(U_\alpha \cap U_\beta \cap U_\gamma) \;\rightrightarrows\; \coprod_{\beta<\alpha} G(U_\alpha \cap U_\beta) \right). $$
For limit $\alpha$, we of course define $H_\alpha$ to be the colimit. It is easy to verify that each $H_\alpha$ is the colimit as asserted, and thus $\mathrm{colim}_{\alpha<\lambda} \; H_\alpha = \mathrm{colim} \; G$.

Now suppose $G$ has the property that the map $K_\alpha \to G(U_\alpha)$ is injective on objects (a cofibration) for every $\alpha$. Then the pushout defining $H_{\alpha+1}$ is a homotopy pushout and thus homotopy-invariant. Moreover, the map $H_\alpha \to H_{\alpha+1}$ is again a cofibration, so the colimits at limit stages are also homotopy colimits and thus homotopy-invariant. Therefore, for $G$ with this property, strict colimits are 2-colimits. But our functor $c\mapsto \Pi_1(\chi(c),A)$ does have this property (it is a slightly stronger version of being "Reedy cofibrant at the top level").

Thus, its strict colimit is also a homotopy colimit, so $\Pi_1(X,A)$ is equivalent to this strict colimit. The theorem of Ronnie and coauthors asserts that it is in fact *isomorphic* to this strict colimit, but it is easy to check that it has the same set of objects as the strict colimit (namely $A$), and an equivalence of groupoids which is bijective on objects is an isomorphism.