A main point I would make is that connected unions of non connected spaces are commonplace, and an example is given on groupoids page which you are welcome to use, of course.
See also arxiv:1404.0556 for a correction to a proof in my book about the Phragmen-Brouwer Property, with a useful new result on pushouts of groupoids.
The virtue of the use of groupoids is not to give a "nice" proof of a result about fundamental groups, as in some expositions, but to be able to deal with many more examples, of a type usually ignored, as trying to use covering space methods gets unwieldy.
I'll add that in a talk I gave in Chicago in 2012 I called the fact that the usual van Kampen theorem for the fundamental group does not compute the fundamental group of the circle, THE basic example in topology, is a real anomaly. This talk, with others, is available on my preprint page.
Edit 25 April: I found the groupoid SvKT strange since in algebraic topology you usually get exact sequences which do not give complete information; yet here one computes $\pi_1(X,A)$ as a colimit which completely determines any group $\pi_1(X,a)$, though to find that you have to do some combinatorial work. A reason for the success seems to be that groupoids have structure in dimensions $0$ and $1$, allowing the modelling of gluing $1$-types.
So the question was: can this principle be extended to higher dimensions, using new algebraic objects with structure in dimensions $0, \ldots,n$, allowing for the fact that identifications in low dimensions have homotopical effect in high dimensions? Thus the $1$-dimensional theorem pointed to potential higher dimensional results, in which the invariants one might want, such as homotopy groups, were part of a much larger structure which was more computable, thus allowing some new computations of old invariants.
September 3, 2014 I give a link here to a presentation I gave at the IHP, Paris, to a workshop on Homotopy Type Theory, on "The intuitions for cubical sets in nonabelian algebraic topology". The talk is also about the notion of composition for homotopically defined structures, and its relations to the origins of algebraic topology. This talk develops some themes in my 2012 Chicago talk.