I'm trying to understand the notion of an accessible category. This isn't the first time I've tried to do this; but every time I try to make sense of the definition, I become perturbed by the following issue: not every small category is accessible.
You can find a definition of accessible category at nLab. In brief a (possibly large) category C is accessible if there's a regular cardinal $\kappa$ such that C has $\kappa$-directed colimits, and that there's a set of $\kappa$-compact objects so that every object C is a $\kappa$-directed colimit of things in this set. Thus, the behavior of the category C is somehow "controlled" by a small subcategory; roughly speaking, all objects of C "look like" filtered colimits of objects in the small subcategory.
Any small category is of course "controlled" by a small subcategory, namely itself, so you'd think that all small categories are accessible. Not quite! The correct statement is:
A small category is accessible if and only if it is idempotent complete
This is proved (I think) in Adamek & Rosicki, Locally Presentable and Accessible Categories. There is a also a proof of an $\infty$-category version of this in Lurie's Higher Topos Theory.
My question is not about the proof of this claim (which I think I understand), but about the underlying motivation for the notion of accessible category. Basically, I'd like one of two things:
Make me understand why it's such a good thing that not every small category is accessible, or
Tell me that "accessible category" is not exactly the right idea, and that there's a generalization of it which includes all small categories a special case.
(Note: the class of accessible categories is closed under a bunch of constructions, such as taking undercategories, or taking functors from a fixed small category. The generalization of 2 ought to have the same properties.)