I am surely not a historian of topology, but I might try a few words.

That the usual literature concerning model categories is quite far away from traditional homotopy as presented in Whitehead's classic, is no wonder. Indeed, model categories are abstracted from homotopy theory, but not really that of the classical flavour. Quillen's lecture notes are not without reason entitled 'Homotopical Algebra'. As discussed in its introduction, its main object is to present an abstract framework where one can consider simplicial objects in categories of relevance for algebra. This leads to a theory of "non-additive derived functors", e.g. André–Quillen homology. 

In particular, the example of the model structure on topological spaces inducing the classical homotopy category is not presented in Quillen's book — only the one using Serre fibrations, generalized CW-complexes and weak homotopy equivalences. The model structure with Hurewicz fibrations/cofibrations and homotopy equivalences had to wait until Strøm's [The homotopy category is a homotopy category][1]. As a consequence, the first absorbers of the theory of model categories were more simplicial minded guys. See for example Bousfield and Kan's *Homotopy limits, completions and localizations*.

One reason, why the notion of a model category is today so omnipresent in algebraic topology is that they provided a very good framework to discuss the homotopy theory of spectra and it was important to work both simplicially and in topological spaces. But the model structure on topological spaces used here was again the Quillen model structure.

I think, it is only in the last years that topologists are caring more again to reunion classical homotopy theory and model categories. One important work for this is Cole's [Mixing model structures][2]. Here, a model structure on topological spaces is constructed, where the weak equivalences are again the weak homotopy equivalences, but fibrations are now the Hurewicz fibrations. This leads to a theory, where the cofibrant objects are all spaces homotopy equivalent to a CW-complex. This model structure interacts rather well with more classical homotopy theory (using Hurewicz cofibrations and so on) as is seen e.g. [here][3] or in (section 8 of) [this][4], which is also used in the five-author paper [Units of ring spectra and Thom spectra][5]. The reason, why the latter needs the connection to more classical homotopy theory is that the theory of $E_\infty$-spaces stems from classical homotopy theory and is simultaneously deeply linked to modern stable homotopy theory.


  [1]: https://doi.org/10.1007/BF01304912 "Arne Strøm, The homotopy category is a homotopy category, Archiv der Mathematik 23 (1972), 435–441"
  [2]: https://doi.org/10.1016/j.topol.2005.02.004 "Michael Cole. Mixing model structures. Topology and its Applications 153(7) (2006), 1016–1032"
  [3]: https://arxiv.org/abs/math/0411656 "J. P. May, J. Sigurdsson. Parametrized homotopy theory. 2004"
  [4]: https://arxiv.org/abs/0706.2874 "Michael A. Shulman. Parametrized spaces model locally constant homotopy sheaves. 2007"
  [5]: https://arxiv.org/abs/0810.4535 "Matthew Ando, Andrew J. Blumberg, David J. Gepner, Michael J. Hopkins, Charles Rezk. Units of ring spectra and Thom spectra. 2009"