It seems that for you "foundational" means concerning the foundation of ordinary (non-set theoretic) mathematics. In this sense, I would claim that this sort of foundational concern has been a very minor aspect of set theory for decades. Moreover, set theory has never served as foundation for mathematics in any deep practical sense. Probably, Homotopy Type Theory will not really serve as foundation for mathematics in any practical sense either.
This really is not due to any particular nature of set theory or any other foundations. The majority of mathematicians do not care about foundations. They care about the ideas and objects they are working with not how in particular these objects are formalized. It quite conceivable that in the future other foundational ideas may appear as a language tool. But most likely, mathematician will just casually throw arount these new terms from their new foundations, just like mathematicians now casually throw out the word "sets", "infinity", "cardinality", "intersection", "functions", etc.
As for my claim that foundation is a very minor concern for most set theorist: I do think there are many set theorist that are currently working on formalizing any branch of natural mathematics in set theory. I think set theory has long evolved from the study of sets in your sense. Set theory is now the study of the combinatorics of infinity and certain logical phenomena. Homotopy type theory or something else may replace set theory as the prominent language for doing ordinary math, but there is nothing currently available that appear to be able to replace this study of infinity.
Probably, this is because set theory and infinity are inseparable. Since ancient times people were vaguely familiar with infinity. The beginning of the systematic study of infinity is one and the same as the birth of set theory. Cantor was probably the first person to mathematically study different forms of infinity and this was precisely when set theory began. Calculus existed independently of set theory. Did the mathematical branch of infinity really even exists before Cantor began set theory?
On the logical side, modern set theory attempts to produce independence and consistency results. No other foundation has quite as many tools (such as forcing and inner models in set theory) to be able to produce these types of results. If you are interested in the philosophical aspect of some theorem and you know it can be formalizes in several foundations, for practical purpose you would probably use the available tools of set theory to analyze it logical content.
On large cardinals, they are indeed used as foundations but not in your sense. They are foundations for descriptive set theory and infinitary combinatorics. In the current state of mathematics, infinity and set theory is the same thing, so I do not think you would consider this aspect as foundation.