I mention that when I was a writing a topology text in the 1960s whose 1st edition was published in 1968 and whose 3rd edition is now "Topology and groupoids" (2006), I got irritated by the fact that one had to make a detour through a bit, or a lot, of covering space theory to obtain the fundamental group of the circle, THE basic example in algebraic topology. So it was great to find that using the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points ($2$ base points for the circle, not much more than $1$ base point) one could do this and much more; so that book moved over to giving a full account of $\pi_1(X,A)$. And the question of the use of groupoids in topology, particular higher homotopy, has for me been rewarding ever since.
My understanding is that in the intervening 44 years no other text mentions $\pi_1(X,A)$! So who is right?
Grothendieck has written: ""Choosing paths for connecting the base points natural to the situation to one among them, and reducing the groupoid to a single group, will then hopelessly destroy the structure and inner symmetries of the situation, and result in a mess of generators and relations no one dares to write down, because everyone feels they won't be of any use whatever, and just confuse the picture rather than clarify it. I have known such perplexity myself a long time ago, namely in Van Kampen type situations, whose only understandable formulation is in terms of (amalgamated sums of) groupoids.""
May 28: I ought to explain in some detail how groupoids work for $\pi_1(S^1)$. Consider the following two pushout diagrams

where $\mathbf I\cong \pi_1([0,1], \{0,1\})$ is the groupoid with $2$ objects $0,1$ and non-identity arrows $\iota: 0 \to 1, \iota^{-1}: 1 \to 0$; the proof of this is easy since the unit interval is convex in $\mathbb R$. The first diagram shows that the circle $S^1$ is obtained from the unit interval $[0,1]$ by identifying, in the category of spaces, the two end points $0,1$. The second diagram shows that the additive group of integers is obtained from the groupoid $\mathbf I$ by identifying, in the category of groupoids, the two end points $0,1$. Note also that the groupoid $\mathbf I$ has only $4$ arrows, so it is easy to work out everything about it! This groupoid plays a role in the category of groupoids analogous to that of the integers in the category of groups.
All this seems a convincing reason for the result on $\pi_1(S^1)$, to be placed alongside the covering space proof, which is valuable for other reasons.
The above proof is also valuable for suggesting higher dimensional versions, giving, for example, $\pi_n(S^n), \pi_n(S^n \vee S^1)$.