In a prior posting here, the following is a quote from an answer:
$\mathsf{ZFC}$ is so ridiculously overpowered that its inconsistency wouldn't really spill over to the rest of mathematics too much. For the vast majority of mathematical practice, galactically weaker theories like $\mathsf{ZC}$ are enough.
, and it continues to say:
It turns out that a huge amount of mathematics can be developed in $\Pi^1_1 \mathsf{CA_0}$, which is a tiny fragment of $\mathsf{Z_2}$ which is itself a tiny fragment of $\mathsf{ZC}$
My question:
If $\mathsf{ZFC}$ is so ridiculously overpowered (over the goal of formulating mathematics), then why its largely held as the foundation of mathematics? Why should I build a skyscraper just for me to enjoy the shade at summer?
Why not $\mathsf{Z_2}$, or Pocket set theory, or even going high up to MacLane set theory? That seems to be more reasonable. Then we can consider further extensions of those only on demand i.e. when we need to formulate the very scantly amount of mathematics that requires strong foundations, and that would never end.
I think that, from a modern point of view, there is a misunderstanding in the position that you suggest in your question. Really, "set theory" should be understood as an umbrella term that covers a whole hierarchy of ZFC-related theories.
Perhaps one of the most significant advances in foundations is the identification of the consistency strength hierarchy. It allows us to calibrate mathematical statements with "canonical" extensions (guided by large cardinal axioms) or restrictions of ZFC. The former has been of greater interest to set theorists, so it is more visible, but the restrictions are just as important. They allow us to calibrate all sorts of things, from fragments of definable determinacy, to questions of analysis or bounded arithmetic.
In practice, many mathematicians (including many set theorists) only work in one of these restrictions, a small (or even very small) fragment of ZFC. If such a fragment suffices for your purposes, that's the one for you to use. In fact, I would argue that you should strive to work in the fragment that is appropriate for your mathematical needs, using only additional axioms beyond that fragment as your mathematics demands it.
On the other hand, some of us are interested in questions that demand significant strength beyond ZFC. For us, the appropriate theory to work on is an extension of ZFC (or NBG, or MK) with large cardinals, or a theory equiconsistent with such an extension, or a theory expected to be equiconsistent with such an extension. But also, sometimes the questions we want to look at are essentially combinatorial and $\mathsf Z_2$ (or an even weaker fragment) is the right framework in that case.
What matters is that we can go as high or as low within the ladder of the consistency strength hierarchy as the mathematical problems we encounter demand us without having to keep switching frames, so that the set-theoretic scaffolding is in many cases in the background, and only remarked upon when its presence is relevant. (For instance, questions about generic absoluteness demand constant attention to the set-theoretic framework. Questions about the partition calculus may only require a working knowledge of it.)