I was mulling on this previous question of mine, and I think I'd like to play the devil's advocate a bit more. I am now convinced that skeleta do not make category theory any simpler, and this is mostly due to the fact that there is no canonical way to construct a skeleton for a given category.

I am now wondering whether this can be made into a precise theorem. I was thinking of something about this lines (this may not be the right formulation):

Let $Cat$ be the $2$-category of small categories. There is no pseudofunctor $S \colon Cat \to Cat$ such that for every $C \in Cat$, $S(C)$ is a skeletal category equivalent to $C$.

Is this true? I have no clue how to prove this.

And if not, is there some variation which puts the intuition that skeleta involve arbitrary choices on a sound basis?