Set theoretic axioms are used in a few other areas, which are perhaps closely related to set theory but not, in my opinion, "pure set theory".
One area is in "set theoretic topology", which is a field of point-set topology that considers properties that are more sensitive to set theoretic axioms. One example is Fleissner's 1974 paper "Normal Moore Spaces in the Constructible Universe" (Proc. AMS 46:2) which made progress toward the normal Moore space conjecture under the assumption $V=L$. Similarly, Martin's axiom has applications in general topology. For one recent example, see this abstract from the 2018 Joint Math Meetings. Overall, these uses of $V=L$ or Martin's axiom are vaguely analogous to the use of the unproved Riemann hypothesis to obtain results in number theory.
Another branch of math that sometimes uses axioms outside ZFC is set-theoretic combinatorics, although this is probably closer to "pure set theory".
But why is ZFC so heavily used? One reason is certainly sociological: ZFC is heavily used because it is what most mathematicians agree to. But there are two important caveats to that. The first is that most mathematics requires much less than ZFC, so saying "this is provable in ZFC" is usually a vast overestimate of what is needed. In this sense, ZFC is used because it is a safe estimate.
The second caveat is that most mathematicians don't worry about foundations too much, and probably can't name the axioms of ZFC, so for them the use of ZFC to formalize their work is entirely hypothetical. The same goes for any foundational system - similarly few mathematicians could give a reasonable set of definitions for any formal foundational system, be it ZFC or a topos-theoretical system.
As for the axiom of choice, essentially the only place it is not accepted is in certain (but not all) branches of constructive (intuitionistic, non-classical) mathematics. Essentially every contemporary intro textbook to algebra or analysis uses AC when needed to obtain the standard, classical results.
Finally, why don't we assume $V=L$ all the time, to simplify things? The best answer is that, in a sense, the axioms of ZFC are agreed to match the usual conception of "set" (that is, "pure well founded set"), while there is no generally accepted argument that $V=L$ is implicit in the notion of "set". It is always possible that additional axioms could be accepted in the future, if enough mathematicians believed they were natural. But somehow, in practice, the ZFC axioms seem to be the limit of what the term "set" means to the majority of mathematicians who have looked at these things. Much more was written about this in a series of articles "Does mathematics need new axioms" in the December 2000 Bulletin of Symbolic Logic. Separate articles were written by Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel giving different opinions about this exact issue.