It was Euclid's construction, not Leibniz's dissatisfaction with
it, I had in mind as an example of taking a raw equivalence class to be
the "object". As I wrote,
... This suggests that for Euclid "a geometric ratio" was a class of
"equal ratios". . .
So I'm wondering if this answers your question.
You may have been asking a question about the history of the use of
"set theory", so that strictly speaking we do not have "sets" as themata
in Euclid.
I think you missed a point, too, about set theoretic reduction.
It really is logical sloppiness to set theoretically represent a non-set
entity by a set. Call it "set theoretic reductionism" if you like, but
that doesn't excuse it. I see your question as being about when
mathematicians lost their fastidiousness and decided that it was okay to be
sloppy because they knew how not to make any silly "mistakes".
At the same time, what does replace the act of abstraction is the
derivation from the representing sets of the axioms or principles or
theorems of the subject under reduction to set theory. Then, once those
theorems are derived in set theory, THEY ARE WRITTEN OUT IN THE ORDINARY
MATHEMATICAL LANGUAGE, which suppresses all of the strange freight the
familiar mathematical objects being represented by sets has imported.