Interesting question, and possible resolution of the Russell Paradox. This is really in the nature of a comment than an answer.
One can argue that an object is never found without a context; following this thought through gets you Type Theory.
One can argue that objects in themselves are unknowable and thus it is the relationship between that is knowable. This gives Category Theory.
One can argue that inconsistency is rather the rule than the exception; and any reasonable mathematics should not 'explode' on being acquainted with it. This gives inconsistent set theory. It also has another solution to the Russell Paradox: The universal set (the set of everything) exists and is fact the union of all the members of the Russell Set.
This suggests, as you're undoubtably aware, that a good formalisation of Geldsetzers ideas on a different kind of Set Theory may give some very interesting results.
I'm no set theory expert - but the only place I can see in traditional ZFC where sets of sets are introduced is through the power set axiom. There is a paper by Gitman, Hamkins & Johnstone which specifically addresses this - what is the theory ZFC without the power set axiom. Their abstract states:
We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context...Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory ZFC_, obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach.
Further, in the perspective of category theory, you have the category of inhabited sets Set\0; this is obtained from Set by forgetting the structure of initialness. They say:
but the positive word ‘inhabited’ reminds us that this is the simpler notion, which emptiness is defined as the negation of.
Which I suppose is part of Geldsatzers point.
But also there is the category of Pointed Sets where a specific element of any set is distinguished. This can only happen when this category does not contain the empty set. One might think of this as 'compactifying' sets from the purely mathematical sense, where the compacting element is the additional distinguished point.
Finally, traditional Set Theory is built up from sets & membership, and then builds up the idea of a function. Category theory reverses this development - by starting with sets & function and then building up a notion of membership, that is membership is a secondary consideration and not a primary one. Its a structuralist point of view.
Since membership is not primary, a set is ontologically pure - it is indivisible. One cannot ask what members it contains. Similarly one cannot ask what sets, or sets of sets it contains.
For example the set of primary colours is not included within the set of colours, but one can identify some correspondance (by a function). Its a subtle distinction.
It also allows a ramification of ontology: In ZFC, one has only one type - the set; in NFU, New Foundations with ur-elements one has two types - sets & ur-elements (ur-elements are not sets but are contained in sets). In Category Theory there can be many,many more different types.
In Topos Theory, which is a generalisation of Set Theory one actually may have a set with no elements but is not empty. This is because its logic is intuitionistic. So the idea of emptiness ramifies.
Finally, Parmenides said No-Being is Not - which could be interpreted that in actual fact there is no such thing as the emptiness. Everything is always full of something. From this perspective the empty set is a theoretical construct of certain utility in a formal theory but has nothing of the Real about it. In Platos Heaven is there a Form of the Empty that all emptiness in the world is a model of?