Let me begin quoting W. Tait (lectures on proof theory, pages 4 and 5):

> I believe that what further has to be understood, in order to make
sense of these 'paradoxes' is that the notion of a transfinite number or, equivalently,
of a set of transfinite numbers is an essentially **open-ended** notion:
no matter what principles we introduce to construct sets of numbers, providing
only that these principles are well-defined, we should be able to admit all
numbers obtained by these principles as forming a set, and then proceed on
to construct new numbers. So $\Omega$ cannot be regarded as a well-defined extension:
**we can only reason about it intensionally**, in terms of those principles
for constructing numbers that we have already admitted, leaving open in our reasoning the possibility - in fact, the necessity - of always new principles for
constructing numbers. When this is not understood and $\Omega$
is counted as a
domain in the sense of a well-defined extension, then the so-called paradoxes
force on us a partitioning of well-defined extensions into two categories: sets
and proper classes; and the only explanation of why such an extension should
be a proper class rather than a set would seem to be simply that the assumption
that it is a set leads to contradiction. The paradoxes deserve the name
'paradox' on this view because it offers no explanation of why there should
be this dichotomy of well-defined extensions.


Emphases are mine ($\Omega$ is a reference to "the greatest ordinal"). So, one proposed rough sketch of an answer in the direction given by Tait (of course, there are many other directions in philosophy) is this: 

- The subject matter of set theory is open-ended, therefore set theory must be about an intension, the concept of set, not about a well-defined extension. This intension is open-ended (it is hard to make sense of the oxymoron "open-ended well-defined extension"), and it is the unifying criterion behind the plurality of set theoretical practices. The contemporary criterion can be more or less approximated by $ZFC$, but there can be no definite final stage on the progressive conceptual unification of the set-theoretical practices, as there is a neccessary open-endedness (incompleteness) in this intension.

There are many things to address here, but I will not try to pursue them, not even in outline, as this would lead us to a more hardcore philosophical activity.
As a final remark, there are similar arguments in the history of philosophy which were given many years before Russell. One of the most relevant is Plato's third man argument, in *Parmenides*.


SPECULATIVE ADDENDA:

I think the question "why there should be a dichotomy of well-defined extensions and how can we deal with it?", a natural outcome of this discussion, is very relevant for the foundations of set theory, and there are many hints about this in traditional philosophy, say, from Plato to Hegel. (A small digression: "Platonism", as the term appears in the original question, has probably a very weak connection to Plato.  Plato is very subtle, he wrote dialogues, not theoretical treatises in philosophy, in which the dramatic elements interact with the philosophical elements, probably because he sees philosophy as an argumentative activity he shows in the dialogues, not as a body of theory. See W. Tait, Truth an Proof: The Platonism of Mathematics. Anyway, I think, along with Tait, that the man deserves a better fate.) 

I will not dare to say much more about our questions here, but I would like to remark on the idea that there can be no final conceptual unification, for any unification is synthetic, that is, made on the basis of a new conceptual synthesis which, as "new", cannot be among those very things now unified. If reason operates this way, progressively unifying its previous practices through conceptual synthesis, open-endendness is its fate, and I believe mathematics is the primary example of this.