TITLE: Do we still need model categories?
QUESTION [76 upvotes]: One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak equivalences; better, if you have a simplicial model category, is to take the homotopy coherent nerve of the fibrant-cofibrant objects). 
What other functions, then, do  model categories serve today? I understand that getting the theory of $\infty$-categories off the ground (as in HTT, for instance) requires a significant use of a plethora of model structures. However, if we assume that there exists a good model of $(\infty, 1)$-categories that satisfies the properties we want (e.g. that mapping sets are replaced with mapping spaces, limits and colimits are determined by homotopy limits of spaces), how are model categories useful?
I suppose one example would be computing hom-spaces: a simplicial model category gives you a nice way of finding the mapping space between two objects in the associated localization. However, in practice one only considers cofibrant or fibrant objects in the $\infty$-category in the first place, as in Lurie's construction of the derived $\infty$-category (basically, one considers the category of projective complexes -- for the bounded-above case, anyway -- and makes that into a simplicial category, and then takes the homotopy coherent nerve). 
One example where having a model structure seems to buy something is the theorem that $E_\infty$ ring spectra can be modeled by 1-categorical commutative algebras in an appropriate monoidal model category of spectra (in DAG 2 there is a general result to this effect), and that you can straighten things out to avoid coherence homotopies. I don't really know anything about $E_\infty$-ring spectra, but I'm not sure how helpful this is when one has a good theory of monoidal objects in $\infty$-categories.

REPLY [23 votes]: 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. 
Jan 21, 2017 Parts of the above mentioned presentations have been expanded into  this paper;  it  also discusses  the increasing use of cubical sets with connections, which have been little used so far in theories such as quasi categories, but which in some areas have advantages over simplicial sets.