A special case of the question of whether categories should be skeletal is whether you should pick bases for vector spaces. This was the topic of a previous MO questionthe topic of a previous MO question. After all, the standard skeletal form of the category $\text{Vect}(k)_{<\infty}$ (finite-dimensional vector spaces over $k$) is the collection of all $k^n$. In the first pass through linear algebra, students are taught this skeletal model of this category, and then in later iterations they are taught different models.
Taking bases of vector spaces as an example, the skeletal form of a category is both always useful and never useful. One the one hand, abstract arguments are almost always clearer and less error-prone without a skeletal assumption. As I explained in the other question, if you do not push objects into the skeleton of a category, they give you "data types", so that two sides of an incorrect equation often don't even have the same type, instead of merely different values. On the other hand, the skeletal form of a category shows you where the actual numerical information is.
To give another example, if you study semisimple tensor categories, then demanding a skeleton is a bad idea, because it clutters proofs with no gain. But it is also a good idea, because it shows you that the actual data in such a tensor category lies in its $6j$-symbol, or if you like its associator or its tetrahedron symbol.
