One of main reasons that the theory of derivators was introduced is to fix the non-functoriality of the cone construction of triangulated categories. I know that today derivator theory is broad and deep but so far I have not seen (I am not an expert in the field) any concrete application of derivators to the cone construction.

Let's get started with something concrete. In this post, or more precisely, the citation says that a triangulated equipped with a cone functor has to be split. So the basic idea of derivator is that the cone functor, if exists, should come from something higher, and that's why we have to consider diagrams. Therefore we should about diagrams of schemes. Let $\mathbb{D}$ be a triangulated derivator in the sense of [Ayoub's thesis, Definition 2.1.34], and $I$ a small category, $i$ an objects viewed as a morphism $i \longrightarrow I$ and $\mathbf{e}$ the trivial category (one object, one morphism).

Then for every $A \in \mathbb{D}(I)$, and morphism $i \longrightarrow j$ (viewed as a functor over $I$) we have a morphism $j^*A \longrightarrow i^*A$ in $\mathbb{D}(\mathbf{e})$. In this way, we have a $I$-skeleton functor $\mathbb{D}(I) \longrightarrow \mathrm{Func}(I^{op},\mathbb{D}(\mathbf{e}))$.

In particular, if $\mathbf{1}$ is the interval category (the poset $0<1$ viewed as a small category), then there exists a cone functor $\mathrm{Cone}:\mathbb{D}(\mathbf{1}) \longrightarrow \mathbb{D}(\mathbf{e})$ such that for every $A \in \mathbb{D}(\mathbf{1})$, there is a distinguished triangle $$1^*A \longrightarrow 0^*A \longrightarrow \mathrm{Cone}(A) \longrightarrow +1$$ J.Ayoub in some way, uses this fundamental triangle to solve a problem in his thesis: given a distinguished triangle $f^{'} \longrightarrow f \longrightarrow f^{''} \longrightarrow +1$ (distinguished in the sense objectwise), then when does the sequence of right adjoints $g^{''} \longrightarrow g \longrightarrow g'$ be distinguished? I myself see this is a very interesting application and brilliant (I do not if it already appreared in somewhere else or original) and I wonder whether there are many such "simple" applications of derivators?

One of them that I can found is [F. Ivorra + S. Morel, The four operations on perverse motives, Lemma 3.4]. Basically, It states that, fix a derivator (for instance, the derivator $\mathbf{SH}(-)$ motivic stable homotopy category of Morel-Voevodsky-Ayoub) defined over the category of diagrams of $k$-varieties ($k$: a field, but can be replaced by any scheme), the objectwise cone of the unit morphism $\mathrm{id} \longrightarrow f_*f^*$ of a morphism of scheme is a functor. It is a one line proof.

The common point of these two examples is that it gives us some criteria for which a objectwise cone of a natural transformation between triangulated functors can be turned into a functor. Again, I am demanding of such "simple", interesting applications, I think there "should" have many (as it is the motivated reason of the theory) but I cannot find them myself. Thank you in advance.