I don't completely understand the question, because I don't generally see a problem with locally small ordinary categories. My own experience has generally been that for some esoteric things category theorists want to do, it may seem helpful to be able to form e.g. functor categories between large categories, but usually there are other ways to do the same thing, and for practical applications these issues almost never arise.

However, as a sociological observation, I think there are a couple of reasons why local smallness conditions are more ubiquitous in enriched category theory from the perspective of a category theorist working abstractly.

One is that if you're working in generality with an arbitrary enriching category $\cal V$, there is no canonical choice of an enlargement $\cal V^+$. You can produce such enlargements by Yoneda-embedding $\cal V$ into $\rm Set^+$ and perhaps cutting down to some nice subcategory, as Kelly describes in sections 3.11-3.12 of Basic concepts of enriched category theory, but this is still noncanonical and it feels weird to be talking about these presheaves when what you "really want" to be enriching over is the $\cal V$ that you were given. In particular cases, there is usually a natural choice of a $\cal V^+$ such as your $\rm Ab^+$, but these are different from any of the $\cal V^+$s that arise from the general theory. There are general classes of $\cal V$ that admit canonical enlargements of this sort, such as locally presentable categories, but even that doesn't include all the natural enlargements, e.g. $\rm Top^+$ consisting of "collections with a topology". So it's just cleaner to formulate things as much as possible to avoid having to introduce a $\cal V^+$ at all.

Another is that universal properties in enriched category theory are generally weighted, involving weight functors such as $\Phi : \cal C \to \cal V$ landing in the enriching category. And here you really do want it to land in $\cal V$ rather than $\cal V^+$. When talking about small limits and colimits, as we usually do, of course the weight landing in $\cal V$ is part of this smallness. But even for "large" limits and colimits, such as arise for instance in the consideration of totally cocomplete categories, you usually want the weights to be $\cal V$-valued. Again, you could add explicit hypotheses stating this all the time, but it's just cleaner to avoid introducing $\cal V^+$ in the first place.

Perhaps these considerations don't apply so much to "working" mathematicians such as algebraists and geometers who are concerned with specific enrichments such as Ab-categories. But here another consideration comes into play: those mathematicians don't want to have to think about changing universe levels all the time. Most of them probably don't fully understand set-theoretic issues and what you can and can't do when passing back and forth between universes, and so the idea makes them rather uncomfortable, when all they want to do is use additive categories for a practical purpose and all the additive categories they actually encounter are locally small.

Of course those considerations apply to ordinary categories too. And, indeed, many people do define ordinary categories to always be locally small. But some don't, and I think the reason is just what's easy. When defining a category, if you're using the "one collection of morphisms" definition, then the collection of morphisms has to be large no matter what, and so it's easier to just say that; local smallness then has to be added as an extra assumption, so it seems natural to give it a name rather than including it in the definition of "category". If you're using the "many collections of morphisms" definition, then either way is equally easy; you can say that each $\hom(X,Y)$ is a set or is a collection, depending on taste. But for enriched categories two things are different. First, you basically have to use the "many collections of morphisms" definition, so allowing local-largeness has to be a choice rather than being the thing you get naturally. And second, every mathematician knows what an abelian group is, and it's defined to have a set of elements; so the most natural thing to write down when defining an additive category is that "each $\hom(X,Y)$ is an abelian group" which then automatically and implicitly yields local smallness; it would be extra effort to say "each $\hom(X,Y)$ is a collection with an operation satisfying the axioms of an abelian group", and no working mathematician wants to do that when they don't see any benefit from it.