To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$.
On the other hand one can (at least for discrete groups) easily construct the delooping groupoid which is a one-object groupoid with morphisms given by the elements of $G$ together with the induced composition laws.
According to the people at nlab these two constructions should be equivalent under the homotopy hypothesis. Except by just formally invoking this theorem, I am not really able to see how these are equivalent. Is there a concrete interpretation of this statement and accordingly a clear way to show this?
So after reading the paper "Classifying spaces and Spectral sequences" by G. Segal in more detail, the conclusion seems to be the following:
Let $\mathcal{G}$ denote the (delooping) groupoid obtained from $G$ in the way I wrote in the question. For this groupoid one can construct the simplicial nerve $N\mathcal{G}$ which by geometric realization gives us a topological space $B\mathcal{G}$. Then it turns out that in some cases this space gives us the base space of a universal bundle and hence a classifying space for $G$.
This space is however always related to the classifying space $BG$, as constructed by Milnor, by collapsing degenerate simplices. So it always seems to work out (no restrictions necessary on $G$).
Remark: The space $BG$ given by the Milnor construction can also be obtained through the geometric realization process.