Your question does not seemed aimed at set theorists, but let me give a set theorist's answer.

I view the set/class distinction as analogous to and ultimately no more problematic really than the other distinctions of size that are commonly made in mathematics.

For example, we study the finite groups as a robust, coherent collection, and we are untroubled by the fact that there are many than finitely many isomorphism types. We just don't find it confusing that there are infinitely many finite groups. (For example, we don't expect to deduce by Zorn's lemma that there are maximal finite groups.) Or we study the collection of countable graphs, while realizing that there are uncountably many instances even on the same set of vertices. More generally, we might look at $\kappa$-dense topological spaces, or at all structures of a given type of size less than a cardinal $\kappa$, or at spaces of a given dimension or rank, and so on.

These distinctions of size are extremely common and part of the way that we think mathematically; these distinctions are part of the way that we carve up our mathematical universe at its joints. Similarly, we may handle the set/class distinction, which is of the same character, neither especially mysterious or problematic.

In each case, we have to pay attention to the details of the mathematical constructions that we employ, in order that these constructions not take us out of the class in focus.

As you say, set theory is replete with these considerations of size and similar distinctions. The entire large cardinal hierarchy is an investigation of different sizes of infinity. The Grothendieck universe concept, arising at the entryway of that hierarchy, is a such measure of size distinction, usually considered a bit crude or clumsy by set theorists, but useful for non-set-theorists because it is easy to understand. Meanwhile, set theory is full of other subtler universe concepts: the levels of the arithmetical and projective hierarchies provide "universes" of complexity for countable objects; the various cut-off universes $H_\kappa$, $L_\kappa$, $V_\kappa$ are often used as local universe concepts; the proper-class sized inner models $L$, $\text{HOD}$, $L(\mathbb{R})$, $L[0^\sharp]$ and so on provide limitations of the background universe that is not just of "size", but of set-theoretic complexity. In broad strokes, all these limitations affect mathematical argument in a similar way, since one must pay attention to which kinds of constructions might take you beyond the limitation that has been set.

The set/class distinction is just one more such distinction.