TITLE: How would set theory research be affected by using ETCS instead of ZFC?
QUESTION [33 upvotes]: In "Rethinking Set Theory", Tom Leinster argues in favor of teaching axiomatic set theory via Lawvere's Elementary Theory of the Category of Sets with 10 axioms (but phrased in a way that requires no knowledge of category theory) which uses sets and functions as primitive elements compared to ZFC which uses sets and elements as primitives. (although if you really insist you can reintroduce elements as primitives at the expense of more clauses). This is weaker than ZFC but can be made equivalent to ZFC (or equiconsistent according to François G. Dorais) with the addition of a hardly ever needed 11th axiom.
http://golem.ph.utexas.edu/category/2012/12/rethinking_set_theory.html
How would set theory research be affected by using ETCS instead of ZFC?
Is ETCS less cumbersome than ZFC or more, or does it not matter? Does it make proofs longer/shorter or easier/harder?
Would automated theorem provers work better with ETCS?
To clarify, the question is not about the strength of ETCS: assume whatever is needed to bring ETCS up to strength with ZFC. The question is about practicalities of working with the ETCS axioms. I guess most mathematicians don't directly work with axioms of set-theory at all. So the question is: if your work does involve working directly with such axioms then what difference (if any) does it practically make to you as a set-theory researcher to use a different set of axioms that have equal strength. If I understand correctly the goal of Tom Leinster's paper was to reflect actual (non-set-theoretical)-mathematical practice but my question is about what set-theorists would do if they used these axioms. Assume also that for the purposes of this question ETCS refers specifically to the rephrasing used in this paper that is free of category and topos terminology. Perhaps this rephrasing should be renamed ETS. Adding replacement would give the name ETSR.

REPLY [8 votes]: No one has really addressed the question about automated theorem provers.  I'm going to take this to refer more generally to computer-assisted mathematics, since fully automated theorem provers are still of very limited usefulness to a mathematician.  There are a lot of good arguments that the best foundational system for these is neither ZFC (or its relatives) nor ETCS (or its relatives), but some variety of type theory.
This is mainly because type theory is so closely connected with programming and has good computation behavior, thus meshing very well with computers.  Since the question is not about type theory, I won't say much about it; I just want to point out that ETCS is more like type theory than ZFC is.  So although I kind of doubt that actually using ETCS in a computer proof assistant would be much better than using ZFC, learning to think in ETCS and do mathematics therein will probably stand you in better stead when you try to use a proof assistant based on type theory.