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](http://cantorsattic.info)
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.