A category $mathcal{C}$ consists of pair of classes $(mathcal{C}_0, mathcal{C}_1)$, along with maps $$mathcal{C}_1times_{mathcal{C}_0}mathcal{C}_1rightarrow
mathcal{C}_1rightrightarrows mathcal{C}_0rightarrowmathcal{C}_1.$$
A category is called small, locally small, large, depending on the size of the class $mathcal{C}_1$.

If $mathcal{C}_1$ is a set (which would imply $mathcal{C}_0$ is a set), then, $mathcal{C}$ is called a small category.

If $mathcal{C}_1$ is a proper class, but $mathcal{C}(a,b)$ is a set for each $a,bin mathcal{C}_0$, then, $mathcal{C}$ is called a locally small category.

Anything that does not fall in above two cases is called a large category.
A $2$category is described to be something, that comes with a class $mathcal{C}_0$, a category $mathcal{C}(a,b)$ for each $a,bin mathcal{C}_0$, some more data satisfying some conditions.
If I start with a category that is not locally small, I know that, $mathcal{C}(a,b)$ is not a set. Is it then a “better idea” to consider it as a category, along with some compatibility conditions, instead of “feeling bad” that it is not a set?
Is this how the notion of a $2$category came into picture?
Is the size issue, in any way, related to the necessity of introducing the notion of a $2$category?
Does the same justification hold in introducing the notion of an $n$category?
If none of the above has a fairly positive answer, is there a way to introduce the notion of a $2$category as an outcome of trying to fix the size issue in ordinary category theory? Is there a way to introduce the notion of a $n+1$category as an outcome of trying to fix the size issue in an $n$category?