Higgins' downloadable book Categories and groupoids has quite a lot on computing colimits of groupoids. The point is that the groupoid van Kampen theorem has the probably optimal theorem of this type in
R. Brown and A. Razak, ``A van Kampen theorem for unions of
non-connected spaces'', Archiv. Math. 42 (1984) 85-88.
This involves the fundamental groupoid $\pi_1(X,A)$ on a set of base points, and for this and a general open cover, one needs that $A$ meets each path-component of each $1$-, $2$-, $3$-fold intersection of sets of the cover. This answers a particular point in the question. The case $A=X$ is the theorem as stated in the question, and the special case when $A$ is a singleton is in Hatcher's book. The reduction to $3$-fold intersections essentially relies on the idea of Lebesgue covering dimension.
This result translates the problem from topology into algebra; a particular fundamental group, if one wants it, is kind of hidden in the middle of this colimit of groupoids. One then has to do various combinatorial things such as choosing trees in components of graphs, to find the fundamental group. These methods are directly related to methods of use in combinatorial group theory, so one should think of them, including the notion of covering morphism of groupoids used in Higgins' book, as a form of combinatorial groupoid theory. HNN extensions of groups can also be seen as pushouts of groupoids.
One of the useful tools in Higgins' book is the following: given a groupoid $G$ with object set $X$ and a function $f:X \to Y$ construct a groupoid $U_f(G)$ with object set $Y$. This construction yields free groups, and free products of groups, as special cases. In Chapter 9 of Topology and Groupoids this construction is related to making identifications on a discrete subset of a topological space.
These ideas usefully generalise to higher dimensions, via Higher Homotopy Seifert-van Kampen Theorems: see for example Part I of Nonabelian Algebraic Topology for results on second relative homotopy groups. There is more to be said ..........
Later: I realise I did not answer the question as to the purpose of the generalisation. As suggested, the immediate purpose was to have a theorem which yielded the fundamental group of the circle, which is, after all, THE basic example in algebraic topology. It also gave easily additional examples: for example, let $X$ be the space formed by identifying all corresponding points of two copies of the interval $[-1,1]$ except for the point $0$. Thus $X$ is a non Hausdorff space. From the groupoid version with $A$ consisting of the two points $\pm 1/2$, we obtain that the fundamental group of $X$ is the integers.
As another example, consider the following union of two open sets:
Here one chooses one basepoint in each component of the intersection, in this case $4$ basepoints. This kind of example was in van Kampen's original paper, with no indication of proof. One might also have a connected union of 24 open sets, each with $5$ path components, and the intersections having hundreds of path components. Situations like this arise in combinatorial group theory. In general, one chooses the set of base points according to the geometry of the given situation.
The proof given in the paper referred to is by verification of the universal property, and does not require knowledge that the category of groupoids admits colimits, nor any specific method of constructing them.
For me, this work led to the impression that ALL of $1$-dimensional homotopy theory was better expressed in terms of groupoids rather than groups.
The proof also has the advantage of generalising to higher dimensions, once one has the appropriate higher dimensional homotopical gadgets. (It did take me 9 years to find, in conversation with Philip Higgins, the right $2$-dimensional gadget.)