Here is an interesting example where groupoids are useful. The mapping class group $\Gamma_{g,n}$ is the group of isotopy classes of orientation preserving diffeomorphisms of a surface of genus $g$ with $n$ distinct marked points (labelled 1 through n). The classifying space $B\Gamma_{g,n}$ is rational homology equivalent to the (coarse) moduli space $\mathcal{M}_{g,n}$ of complex curves of genus $g$ with $n$ marked points (and if you are willing to talk about the moduli orbifold or stack, then it is actually a homotopy equivalence)
The symmetric group $\Sigma_n$ acts on $\mathcal{M}_{g,n}$ by permuting the labels of the marked points.
Question: How do we describe the corresponding action of the symmetric group on the classifying space $B\Gamma_{g,n}$?
It is possible to see $\Sigma_n$ as acting by outer automorphisms on the mapping class group. I suppose that one could probably build an action on the classifying space directly from this, but here is a much nicer way to handle the problem.
The group $\Gamma_{g,n}$ can be identified with the orbifold fundamental group of the moduli space. Let's replace it with a fundamental groupoid. Fix a surface $S$ with $n$ distinguished points, and take the groupoid where objects are labellings of the distinguished points by 1 through n, and morphisms are isotopy classes of diffeomorphisms that respect the labellings (i.e., sending the point labelled $i$ in the first labelling to the point labelled $i$ in the second labelling).
Clearly this groupoid is equivalent to the original mapping class group, so its classifying space is homotopy equivalent. But now we have an honest action of the symmetric group by permuting the labels on the distinguished points of $S$.
I agree with the comments above: being a pushout is a categorical property. What is useful is to be able to compute explictly such pushouts and, as you say, free/amalgamated products do so in the category of groups.
In his paper Le théorème de Van Kampen (Cahiers de Topologie et Géométrie Différentielle Catégoriques, 33 no. 3 (1992), p. 237-251. Available on Numdam, http://www.numdam.org/item?id=CTGDC_1992__33_3_237_0), André Gramain gives (part of) an explicit recipe to compute the isotropy groups of a coequalizer of a pair $(\phi,\psi)$ of morphisms of groupoids. This recipe applies to your case by considering (as in van Kampen's theory) the disjoint sum of the groupoids $\pi_1(X_1,A)$ and $\pi_1(X_2,A)$ and the two morphisms from $\pi_1(X_0,A)$ to this disjoint union.
In SGA 1 (Revêtements étales et groupe fondamental, Exposé IX, §5),
Grothendieck had given the same recipe for the fundamental group of schemes. However, his
proof is more categorical and based on the correspondence between coverings and sets with action of the fundamental groups, and on descent theory for coverings.
Best Answer
I can only point to the place where this was originally done (or rather, the latest edition thereof):
Topology and Groupoids by Ronnie Brown
It's a fantastic textbook and easy to read (and cheap, if you buy the electronic copy - the best £5 I've spent). Ideally what you'd do is calculate the equivalent subgroupoid $\Pi_1(S^1,\{a,b,c\})$ where $a,b,c$ are three points in $S^1$, one in each intersection of opens.