The fundamental groupoid and a pushout in the category of groupoids.

You mention Ronnie Brown, but have you looked up his book on topology and groupoids. I think what you ask for is there. The other very early source for this sort if calculation is in P. J. Higgins little monograph which is a TAC reprint at (http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html). There is a lot of stuff in there which you do not find most other places.

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.

Andre Gramain's 1992 exposition of van Kampen's statement is in fact covered explicitly in "Topology and Groupoids", as it was in the 1988 version of that book; it was just an exercise in the 1968 edition.

The point of this type of exposition is to say that sometimes a group can be better explicitly described in terms of groupoids. A basic example is the group $\mathbb Z$ of integers! This is obtained from the groupoid $\mathcal I$ which has two objects 0,1 and exactly one arrow between them by identifying 0 and 1. This is rather analogous to the way the circle is obtained from the unit interval $[0,1]$ by identifying 0 and 1 !

Another aspect is that sometimes a groupoid is a better object to deal with than a group. For example, a homotopy colimit of a diagram of groups is really a groupoid. This is analogous in topology to taking double mapping cylinders rather than a pushout of maps of CW-complexes.

For more information on the book see http://groupoids.org.uk/topgpds.html

I should also say the Higgins' monograph has results on groups, for example a generalisation of Grusko's theorem, that has not been equalled by other methods.

Another aspect of Topology and groupoids is Chapter 11 on "Orbit spaces, orbit groupoids", which allows some computation of the fundamental groupoid and hence group of an orbit space.