Disclaimer: this is not an answer to the question as I have no explanation for why people don't introduce natural transformations in the way explained in the question, but I am posting this in order to expand a comment I made. The comment was
this is the starting observation to make for introducing simplicial categories as a model for $\infty$--categories
Moreover, I am not a specialist neither of category theory nor of homotopy theory (and a posteriori of higher categories).
The $2$-category of categories
The starting point is that the category $Cat$ of categories is actually a $2$-category. For any to objects (i.e. categories) $\mathcal C$ and $\mathcal D$ we have that $Hom_{Cat}(\mathcal C,\mathcal D)$ is itself a category.
This is very transparent when using the definition $$ Hom_{Cat}(\mathcal C,\mathcal D):=Hom_{t_{\leq0}(Cat)}(\mathcal C\times\Delta^1,\mathcal D)\,, $$ where $\Delta^1=\Box^1=G^1$ is the arrow category $0\to 1$ and $t_{\leq0}(Cat)$ is the underlying $1$-category of $Cat$.
Remark: In general one can see a $2$-category $\mathcal C$ as a simplicial category by replacing the $Hom$-categories by their nerves.
In the case of $Cat$, we see that the $Hom$-categories naturally appear as $1$-truncations of simplicial sets (one can replace here "simplicial" by "cubical" of "globular").
The $3$-category of $2$-categories
Le us now go to natural transformations of (strict) $2$-functors between (strict) $2$-categories. Given two such $2$-functors $F,G:\mathcal C\to\mathcal D$ one can see that a natural transformation $F\Rightarrow G$ is the same as a $2$-functors $$ \phi:\mathcal C\times G^2\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^2$ is the $2$-category with two objects $0$ and $1$ and such that $Hom_{G^2}(0,1)$ is the arrow category $G^1=(0\to 1)$.
Therefore the "set" of $2$-functors is a naturally a $2$-category.
Remark: as before we can then see any $3$-category as a simplicial/cubical/globular category by replacing the $Hom$-$2$-categories by their (simplicial/cubical/globular) nerves.
In the case of $2-Cat$, we see that the $Hom$-$2$-categories naturally appear as $2$-truncations of globular sets.
Simplices, Cubes, and globes
The globe category $G$, the cubical category $\Box$ and the simplicial category $\Delta$ are known to be suitable geometric shape to model higher structures. Simplicial sets are good models for (weak) $\infty$-groupoids. It was proved (by Jardine ... with some improvement by Cisinski if I remember well) that cubical sets also provide a model for (weak) $\infty$-groupoids.
I don't know any reference but I guess that the same holds for globular sets (which are quite more used by people working with automata).
The $(n+1)$-category of $n$-categories
Let me consider the category $n-Cat$ of (strict) $n$-categories. A a natural transformation between (strict) $n$-functor $F,G:\mathcal C\to\mathcal D$ can be written as an $n$-functor $$ \phi:\mathcal C\times G^n\to\mathcal D $$ such that $\phi(-,0)=F$ and $\phi(-,0)=G$, where $G^n$ is the $n$-category with two objects $0$ and $1$ and such that $Hom_{G^n}(0,1)$ is the $(n-1)$-category $G^{n-1}$.
Therefore the "set" of $n$-functors is a naturally a (strict) $n$-category, and thus $n-Cat$ is a (strict) $n+1$-category. It also naturally appears as a $n$-truncation of a globular category.
The advantage of working with simplicial/cubical/globular categories
Working directly with simplicial/cubical/globular categories has the following advantages:
- it does allow to work directly with higher categories without going through an inductive process.
- it allows to deal with weak $(\infty,1)$-categories, as simplicial/cubical/globular are models for weak $\infty$-groupoids (here $(\infty,1)$ stands for "$\infty$-categories such that $n$-arrows for $n\geq2$ are weakly invertible").