A version of the Seifert-van Kampen theorem for not necessarily connected spaces was published by me in 1967, see here, and it uses the fundamental groupoid $\pi_1(X,C)$ on a set $C$ of base points. The usual connectivity condition is replaced for a pushout determined by $U \cup V$ by the condition that $C$ meets each path component of $U,V, U \cap V$. You can ask if this method is "bare hands"!
In a sense, the thesis proposed in 1968, maybe not explicitly, is that all of $1$-dimensional homotopy theory is better modelled by groupoids rather than groups.
The book Topology and Groupoids gives a full account of this theory; it is the 2006 3rd edition of a book published in 1968, 1988. See also this mathoverflow discussion, and this reprinted 1971 book Categories and Groupoids by Philip Higgins. A feature of these books is the groupoid construction $U_f(G)$ with object set $Y$ from a groupoid $G$ and a set function $f: Ob(G) \to Y$. This construction includes free groups and free groupoids, and free products of groups. I'll take this opportunity to advertise a small correction to the proof of the Jordan Curve Theorem in T&G: this proof applies groupoid algebra to unions of non-connected spaces.
Alexander Grothendieck wrote in a letter to me in 1983:
" . .... both the choice of a base point, and the $0$-connectedness assumption, however innocuous they may seem at first sight, seem to me of a very essential nature. To make an analogy, it would be just impossible to work at ease with algebraic varieties, say, if sticking from the outset (as had been customary for a long time) to varieties which are supposed to be connected. Fixing one point, in this respect (which wouldn't have occurred in the context of algebraic geometry) looks still worse, as far as limiting elbow-freedom goes!"
[Here is a link to further post 1970 correspondence of Alexander Grothendieck.]
June 10: I should have said from the start that my work on generalisations of the Seifert-van Kampen Theorem for the fundamental group was motivated by the desire to obtain a result which would also yield the fundamental group of the circle. I first extended a method of nonabelian cohomology due to Olum, and this was published in 1965, see here: it gave the fundamental group of a wedge of circles. To my surprise, I then found that the direct method using the fundamental groupoid on a set of base points was both simpler to prove and was more powerful. The cited 1967 paper on that also refers to a paper of Weinzweig, which uses nerves of covers.
The following diagram is another way of looking at the fundamental group of the circle.

Here the groupoid $\mathsf I$ has two objects $0,1$ and one arrow $\iota: 0 \to 1$, and is easily shown to be isomorphic to $\pi_1([0,1],\{0,1\})$.
Nov 17, 2015: I leave the reader to work out how to modify the second diagram if you replace $\{0,1\}$ by $\{0,1,2\}$, $[0,1]$ by $[0,2]$, $S^1$ by $S^1 \vee S^1$, ..... and ....
February 16, 2017 A recent account of the background to and developments of this work is in this article.
April, 2020 See also my answer to this mathoverflow question on using more than one base point for the Van Kampen Theorem. For the circle, it seems like a Goldilocks situation: one base point is too small; the whole fundamentaal groupoid is too big; but two base points are just right!
More generally, what is wrong with a connected space which is the union of $50$ open sets the intersection of any two of which has at least $200$ pathcomponents? The algebraic theory of groupoids is a natural extension of the theory of groups, available for over half a century.