TITLE: Do we still need models of spectra other than the $\infty$-category $\mathrm{Sp}$?
QUESTION [13 upvotes]: This question asked whether $\mathrm{Sp}$ is convenient in the sense of satisfying (in the $\infty$-categorical sense) a list of desired properties of Lewis in his 1991 paper (see there).
The answer given there is yes, provided one interprets Lewis's desiderata in the setting of $\infty$-categories.
Given that $\mathrm{Sp}$ is better behaved than all other existing models of spectra, are them still needed for the purposes of homotopy theory?

Here $\mathrm{Sp}$ means the $(\infty,1)$-category of spectra, not specifically the quasicategory of spectra.
(@Dmitri Pavlov requested clarification on the meaning of $\infty$-category. I meant quasicategory, but only because of ignorance: I don't know constructions of $\mathrm{Sp}$ other than Lurie's, and would be happy if someone could point me to references on this. Regarding the meaning of “$\infty$-category”, I believe the discussion would be more interesting if we do not restrict to quasicategories. (This should be a comment; see the edit summary))

REPLY [16 votes]: The original question has been answered in the sense that there are people who are confident to prove every statement about spectra they care about without recourse to models or model categories of spectra. (That they might argue with statements only known due to specific manipulations of simplices in the model of quasi-categories is a different question as this question is only about models of spectra and not about the use of models or model categories in general.) More precisely, Dylan has expressed this sentiment in the following sentence. 

I know of no homotopically meaningful statement about spectra that can’t be proven without using model categories of spectra.

I do not see the main purpose of spectra in making homotopically meaningful statements about them. Although I regret that my own work has often been rather far removed from it, I see it as one of the main purposes of spectra to facilitate computations of geometric interest. The things we start with a often not constructed in a particularly homotopical fashion. 
Example 1) The Pontryagin-Thom construction. Here we identify bordism groups of manifolds with the homotopy groups of a Thom spectrum and then use e.g. its homology and the Adams spectral sequence to actually compute these groups. The Adams spectral sequence of course works completely in the homotopy category of spectra, but how do I prove the Pontryagin-Thom theorem if I only know that spectra are the free presentable stable infinity-category on one object? If I am given a problem about topological spaces (even manifolds) it seems to very useful for me that I can use a sequence of pointed topological spaces with suspension maps between them to get a spectrum. 
Another nice thing about models is that you can perform very concrete geometric operations on your geometrically given objects, then check (e.g.) some cofibrancy and see that they coincide with homotopy invariant constructions. As a simple example: Say you are given a concrete topological space with a $G$-action and want to compute the $G$-homotopy orbits (or rather the quotient in the $\infty$-category of spaces). Then I can check in my model sometimes that the action is free and just perform a usual quotient, of which (e.g.) the homology groups are usually much easier to compute than of the homotopy orbits. A recent more sophisticated example of this philosophy is the following:
Example 2) For a topological group $G$, let's denote by $C_n(G)$ the space of commuting $n$-tuples of elements in $G$. In a recent paper, Gritschacher and Hausmann compute the homotopy groups of $C_n(O)$ by identifying the $C_2$-equivariant homotopy type of $C_n(U)$ with $\Omega^{\infty}(k\mathbb{R}\wedge (S^{\sigma})^{\times n})$. They check cofibrancy conditions, write down homeomorphism etc. -- i.e. they argue in a very non-$\infty$-categorical way. But how should one even start calculating the homotopy groups of a very concrete space without arguing first in models to translate it into something that can be described in a purely homotopical fashion?
I want to add that Gritschacher and Hausmann also provide a proof of real Bott periodicity in this fashion (essentially filling in details into a very short sketch of Suslin). Here the case is different as the K-theory spectrum of the symmetrical monoidal category of finite-dimensional real vector spaces has already an $\infty$-categorical sound to it. Is there a proof of real (or complex) Bott periodicity in a purely $\infty$-categorical style starting from the description of K-theory as above? 
Example 3) I think, similar points as above apply to the (equivariant) analysis of the homotopy groups of symmetric powers of spaces. See e.g. Schwede's article

It goes without saying that the $\infty$-categorical paradigm is extremely important and useful, especially for proving anything that can be formulated in purely $\infty$-categorical style - as Dylan said: even if there is a short-cut using models often the $\infty$-categorical argument buys you more. But I see arguing with models of spectra still as important if you start with something that is not given in $\infty$-categorical style, but rather as concrete topological spaces.