Is it a mistake ab initio to think of categories as of models of category theory, just as we think of (inner) models-of-set-theory as of models of set theory, graphs as of models of graph theory, groups as of models of group theory, topologies as of models of topology, and so on?
There is one big difference, and maybe the essence lies within: all "normal" models have to been based on sets, while categories may be based on proper classes.
Why would it be a problem for "normal" models of "normal" theories to be based on proper classes, but not for categories as models of category theory?
Side remark: Forgive me for talking at large: "Normal" models of "normal" theories are required to be based on sets, because only then one can apply the machinery of set theory. For categories one definitely wants to apply another machinery, the machinery of category theory. But if this machinery can handle class models in principle, why not apply this machinery on class models of "normal" theories, too - and in a direct way?