Besides category theory itself, is there any part of mathematics that categories that are not locally small play an important role?
Moreover, I don't understand why people care about size issues of category theory. I feel like category theory is just a type of language that we use when we cheat, i.e. we want to use a single definition to mean lots of things and see what are true under that definition. However, when these ideas are applied to sets, then we don't have to worry about all consistency problems arising due to size issues.
For example, for a given abelian category $\mathscr{A}$, we can construct the category of chain complexes $Kom(\mathscr{A})$ and show that this is an abelian category again. So we can form a finite iteration $Kom(...Kom(\mathscr{A})...)$. With this in one hand, when this is directly applied to $\mathscr{A}=$Ab, we can formulate this purely in the zone of ZFC. So I don't see "embedding category theory into ZFC" is a big deal, since when we apply categorical ideas to sentences formulated just with sets, all proofs can be purely written in ZFC. Am I misunderstanding something?
Categorical language would be useless in its role as a conceptual shorthand in important applications such as algebraic geometry or algebraic topology, if it were restricted to small categories. This is not because such language is cheating but because ZFC is not an adequate practical universal language for mathematics. In the example you give, the category of all abelian groups $\mathbf{Ab}$ does not comprise a set in ZFC and that severely limits what one can conveniently say about it in ZFC. E.g., you can't quantify over all functors from $\mathbf{Ab}$ to itself. You can translate most proofs that use categorical language into "morally equivalent" proofs in ZFC, but the translations will generally involve complications about set-theoretical issues that are irrelevant to the underlying mathematical ideas. ZFC is a great technical tool, but it is not the ultimate right way to formalise mathematics - we haven't discovered that yet $\ddot{\smile}$.