I confess to being confused by all this $(\infty,1)$ category business, and the way $\Pi X$ is used as another name for the singular simplicial set of $X$. This is related to Peter's question on computations.
I thought one reason for moving from loops or paths to fundamental groups or fundamental groupoids, i.e. taking homotopy classes, was that one could do specific computations in groups, and also groupoids. So I began in the 1960s to look for higher dimensional versions of these groupoid methods, again with the aim, or hope, of higher dimensional nonabelian calculations. Of course we were well aware of all the laws on paths, or singular simplices, up to homotopies, e.g. Kan extension conditions, but it seemed difficult to get computational information directly at the path space or singular complex level.
What was surprising, and took a long time to realise, was that we could do these higher dimensional strict groupoid methods, using certain homotopy classes, for certain structured spaces, particularly filtered spaces (11 years), and later $n$-cubes of spaces (Loday) (17 years). In the filtered space work, the insights of model theory have also proved very useful - I think they have not yet been used in the $n$-cube situation. Grothendieck was amazed when I told him in 1985 (6?) that $n$-fold groupoids model homotopy $n$-types (Loday's theorem). Since we can use this idea for specific nonabelian colimit calculations in homotopy theory with the aid of a Higher van Kampen type theorem, I am happy as an old man to rest with the use of strict multiple groupoids of various kinds suitable for the problem at hand. Of course in the proofs, the relations between the weak (spaces of maps) situation and the strict one is crucial.
I see these ideas as another contribution to the tool kit of algebraic topology, and some younger people are using them.
It seems a useful, but not obligatory, test of a theory to ask if it can in some cases produce some numbers not previously available.
February 25, 2015 Here is a link to a talk in Galway, December, 2014, giving further background to this answer, particularly in relation to the history of algebraic topology.
November 8, 2015: Here is a June, 2015, presentation on
A philosophy of modelling and computing homotopy types,
which gives the background to using, for specific computations, some algebraic models of homotopy types in the form of strict higher groupoids. It is the strictness which leads to precise colimit computations in homotopy theory, generalising those well understood for fundamental groups. One construction these ideas have led to is a nonabelian tensor product of groups, whose current bibliography has 138 items, mainly by group theorists.