> it's not clear to me why you necessarily want the language you use to rigorously investigate the foundations of mathematics to be the same one you use to do proof verification.
Among other things, I want theorems about foundations to be useful to non-foundational mathematicians. Working on type theory can improve, at the very least, formal verification tools. In my view, traditional ZFC-based mathematical logic has a much smaller likelihood of being useful for regular mathematicians. For example, the vast majority of mathematicians will never explicitly assume the existence of a large cardinal in a paper. It seems to me that the difference in usefulness is probably due to their relative distance from common discourse.
More generally, it's not clear to me why you necessarily want the language you use to rigorously investigate the foundations of mathematics to be the same one you use to do proof verification. ZFC – and set-theoretic approaches more broadly – facilitate proving things about mathematics (meta-mathematical theorems), and for that they are very useful. E.g., the strength of a theory is equivalent to the size of the cardinals it assumes, loosely speaking [1].
If HoTT helps you prove theorems, great. In particular I could see it being of interest to people working in higher category theory (and such people have responded in other comments here!). My remarks were meant for the other 99% of mathematicians – most of whom go their entire lives without ever writing the word "category" in a paper.